math hands
math hands Calculus: HW set 150c01s04 don’t try to find the answer, try to find meaning and understanding 1. Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. 5 x→∞ x lim
x values
x
expression values
5 x
1000
100, 000
1, 000, 000
100, 000, 000
∞
2. ridiculously famous [e] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x 1 1+ x→∞ x lim
x values expression values
1000
x 1+
100, 000
1, 000, 000
100, 000, 000
∞
1 x x
3. a little famous [e2 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x 2 1+ x→∞ x lim
x values expression values
pg. 1
1000
x 1+
100, 000
1, 000, 000
100, 000, 000
∞
2 x x
math hands
c
2007-2009 MathHands.com v.125
4. a little famous [e−1 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x −1 lim 1 + x→∞ x
x values expression values
1000
x 1+
100, 000
1, 000, 000
100, 000, 000
∞
−1 x x
5. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x1 1 lim x→∞ x
x values expression values
1000
x 1 x
100, 000
1, 000, 000
100, 000, 000
∞
x1
6. super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
x values expression values
1000
x x sin
1 x
1 x
100, 000
1, 000, 000
100, 000, 000
∞
7. super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit.
pg. 2
math hands
c
2007-2009 MathHands.com v.125
1 lim x 1 − cos x→∞ x
x values expression values
1000
x x 1 − cos
1 x
100, 000
1, 000, 000
100, 000, 000
∞
8. spoof on a super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
x values expression values
1000
x x sin
1 2x
1 2x
100, 000
1, 000, 000
100, 000, 000
∞
9. spoof on a super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
x values expression values
1000
x x sin
1 4x
1 4x
100, 000
1, 000, 000
100, 000, 000
∞
10. a little famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x→∞
pg. 3
ln(x) x
math hands
c
2007-2009 MathHands.com v.125
x values
x
expression values
ln(x) x
1000
100, 000
1, 000, 000
100, 000, 000
∞
11. a little famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x→∞
x values
x
expression values
x ln(x)
1000
x ln(x)
100, 000
1, 000, 000
100, 000, 000
∞
12. subtle, easy, hard, & important Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim 5 x→∞
x values
x
expression values
5
1000
100, 000
1, 000, 000
100, 000, 000
∞
13. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. 5 x→∞ x lim
14. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
pg. 4
3x + 7 8−x
math hands
c
2007-2009 MathHands.com v.125
15. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
x−
p x2 + 8x − 5
16. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
x−
p x2 + 10x − 5
17. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
7 8 + 3x
18. not famous Find
sin(x) 5+ x→∞ x lim
19. not famous Find
sin(x) 5+ x→−∞ x lim
20. Find lim (12 sin(x)) x→∞
21. a little famous [ π2 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim tan−1 (x) x→∞
pg. 5
math hands
c
2007-2009 MathHands.com v.125
x values
x
expression values
tan−1 (x)
1000
100, 000
1, 000, 000
100, 000, 000
∞
22. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (.75)x − 1 x→∞ .75 − 1 lim
x values
x
expression values
(.75)x −1 .75−1
20
50
100
200
∞
23. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (.95)x − 1 x→∞ .95 − 1 lim
x values
x
expression values
(.95)x −1 .95−1
20
50
100
200
∞
24. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (1.05)x − 1 x→∞ 1.05 − 1 lim
pg. 6
x values
x
expression values
(1.05)x −1 1.05−1
20
math hands
50
100
200
∞
c
2007-2009 MathHands.com v.125
25. very famous rx − 1 x→∞ r − 1 lim
pg. 7
math hands
c
2007-2009 MathHands.com v.125
math hands Calculus: HW set 150c01s04 don’t try to find the answer, try to find meaning and understanding 1. Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. 5 x→∞ x lim
solution: x values
x
expression values
5 x
1000
100, 000
1, 000, 000
100, 000, 000
0.005
0.00005
0.000005
0.00000005
∞ by EM ≈ 0
Solution:
2. ridiculously famous [e] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x 1 lim 1 + x→∞ x
solution: x values expression values
x 1+
1 x x
1000
100, 000
1, 000, 000
100, 000, 000
2.71692393
2.71826824
2.71828047
2.71828181
∞ by EM ≈ 2.71828
Solution:
3. a little famous [e2 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit.
pg. 1
math hands
c
2007-2009 MathHands.com v.125
x 2 lim 1 + x→∞ x
solution: x values expression values
x 1+
2 x x
1000
100, 000
1, 000, 000
100, 000, 000
7.37431239
7.38890832
7.38904132
7.38905595
∞ by EM ≈ 7.38906
Solution:
4. a little famous [e−1 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x −1 1+ x→∞ x lim
solution: x values expression values
x 1+
−1 x x
1000
100, 000
1, 000, 000
100, 000, 000
0.36769542
0.3678776
0.36787926
0.36787944
∞ by EM ≈ 0.36788
Solution:
5. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x1 1 lim x→∞ x
solution:
pg. 2
math hands
c
2007-2009 MathHands.com v.125
x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
0.99311605
0.99988488
0.99998618
0.99999982
x 1 x
x1
∞ by EM ≈ 1
Solution:
6. super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
1 x
solution: x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
0.999999833333
0.999999999983
1
1
x x sin
1 x
∞ by EM ≈ 1
Solution:
7. super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. 1 lim x 1 − cos x→∞ x
solution: x values expression values
pg. 3
x x 1 − cos
1 x
1000
100, 000
1, 000, 000
100, 000, 000
0.0000005
0.00000000005
0.000000000001
0
math hands
∞ by EM ≈ 0
c
2007-2009 MathHands.com v.125
Solution:
8. spoof on a super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
1 2x
solution: x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
0.499999979167
0.499999999998
0.500000000000
0.500000000000
x x sin
1 2x
∞
by EM ≈ 0.5
Solution:
9. spoof on a super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
1 4x
solution: x values expression values
x x sin
1 4x
1000
100, 000
1, 000, 000
100, 000, 000
0.249999997396
0.250000000000
0.250000000000
0.250000000000
Solution:
pg. 4
math hands
c
2007-2009 MathHands.com v.125
∞
by EM ≈ 0.2
10. a little famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x→∞
ln(x) x
solution: x values
x
1000
100, 000
1, 000, 000
100, 000, 000
expression values
ln(x) x
0.0069078
0.0001151
0.0000138
0.0000002
∞ by EM ≈ 0
Solution:
11. a little famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x→∞
x ln(x)
solution: x values
x
1000
100, 000
1, 000, 000
100, 000, 000
expression values
x ln(x)
144.7648273
8685.8896381
72382.4136505
5428681.0237906
∞ by EM ∞
Solution:
12. subtle, easy, hard, & important Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim 5 x→∞
solution:
pg. 5
math hands
c
2007-2009 MathHands.com v.125
x values
x
1000
100, 000
1, 000, 000
100, 000, 000
expression values
5
5.0000000
5.0000000
5.0000000
5.0000000
∞ by EM ≈ 5
Solution:
13. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. 5 x→∞ x lim
solution: PIM succeeds here. limx→∞
5 x
PIM
=
5 ∞
=0
Solution:
14. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
3x + 7 8−x
solution: PIM does NOT succeed here, since we get use EM for a sensible guess.
∞ −∞ ,
an indeterminate form, thus more work is needed. We
solution: x values expression values
x
3x+7 8−x
1000
100, 000
1, 000, 000
100, 000, 000
-3.03125
-3.00031002
-3.000031
-3.00000031
∞ by EM ≈ −3
Solution:
pg. 6
math hands
c
2007-2009 MathHands.com v.125
15. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
x−
p x2 + 8x − 5
solution:
√ PIM does NOT succeed here, since we get ∞ − ∞2 + 8∞ − 5 = ∞ − ∞, an indeterminate form, thus more work is needed. We use EM for a sensible guess. solution: x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
-3.989541778
-3.999895004
-3.9999895
-3.999999894
x x−
√ x2 + 8x − 5
∞ by EM ≈ −4
Solution:
16. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
x−
p x2 + 10x − 5
solution:
√ PIM does NOT succeed here, since we get ∞ − ∞2 + 10∞ − 5 = ∞ − ∞, an indeterminate form, thus more work is needed. We use EM for a sensible guess. solution: x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
-4.985074516
-4.999850007
-4.999985
-4.999999849
x x−
√ x2 + 10x − 5
Solution:
17. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
pg. 7
7 8 + 3x
math hands
c
2007-2009 MathHands.com v.125
∞ by EM ≈ −5
solution: PIM succeeds here. limx→∞
7 8+3x
PIM
=
7 8+3∞
=
7 8+∞
=
7 ∞
=0
Solution:
18. not famous Find
sin(x) 5+ x→∞ x lim
Solution: hint: graph it
Solution:
19. not famous Find
sin(x) lim 5+ x→−∞ x
Solution: hint: graph it
Solution:
20. Find lim (12 sin(x)) x→∞
Solution: hint: graph it
pg. 8
math hands
c
2007-2009 MathHands.com v.125
Solution:
21. a little famous [ π2 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim tan−1 (x) x→∞
solution: x values
x
1000
100, 000
1, 000, 000
100, 000, 000
expression values
tan−1 (x)
1.56979633
1.57078633
1.57079533
1.57079632
∞ by EM ≈ 1.5708
Solution:
22. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (.75)x − 1 x→∞ .75 − 1 lim
solution: x values
x
20
50
100
200
expression values
(.75)x −1 .75−1
3.98731515
3.99999773
4
4
∞ by EM ≈ 4
Solution:
23. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit.
pg. 9
math hands
c
2007-2009 MathHands.com v.125
(.95)x − 1 x→∞ .95 − 1 lim
solution: x values
x
20
50
100
200
expression values
(.95)x −1 .95−1
12.83028155
18.46110049
19.88158942
19.99929895
∞ by EM ≈ 20
Solution:
24. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (1.05)x − 1 x→∞ 1.05 − 1 lim
solution: x values
x
20
50
100
200
expression values
(1.05)x −1 1.05−1
33.0659541
209.34799572
2610.02515693
345831.6163032
∞ by EM ∞
Solution:
25. very famous rx − 1 x→∞ r − 1 lim
Solution: hint: your answer should begin such as ”depends on r, if r is ......??
pg. 10
math hands
c
2007-2009 MathHands.com v.125
Solution:
pg. 11
math hands
c
2007-2009 MathHands.com v.125
x values
x
expression values
5 x
1000
100, 000
1, 000, 000
100, 000, 000
∞
2. ridiculously famous [e] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x 1 1+ x→∞ x lim
x values expression values
1000
x 1+
100, 000
1, 000, 000
100, 000, 000
∞
1 x x
3. a little famous [e2 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x 2 1+ x→∞ x lim
x values expression values
pg. 1
1000
x 1+
100, 000
1, 000, 000
100, 000, 000
∞
2 x x
math hands
c
2007-2009 MathHands.com v.125
4. a little famous [e−1 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x −1 lim 1 + x→∞ x
x values expression values
1000
x 1+
100, 000
1, 000, 000
100, 000, 000
∞
−1 x x
5. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x1 1 lim x→∞ x
x values expression values
1000
x 1 x
100, 000
1, 000, 000
100, 000, 000
∞
x1
6. super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
x values expression values
1000
x x sin
1 x
1 x
100, 000
1, 000, 000
100, 000, 000
∞
7. super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit.
pg. 2
math hands
c
2007-2009 MathHands.com v.125
1 lim x 1 − cos x→∞ x
x values expression values
1000
x x 1 − cos
1 x
100, 000
1, 000, 000
100, 000, 000
∞
8. spoof on a super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
x values expression values
1000
x x sin
1 2x
1 2x
100, 000
1, 000, 000
100, 000, 000
∞
9. spoof on a super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
x values expression values
1000
x x sin
1 4x
1 4x
100, 000
1, 000, 000
100, 000, 000
∞
10. a little famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x→∞
pg. 3
ln(x) x
math hands
c
2007-2009 MathHands.com v.125
x values
x
expression values
ln(x) x
1000
100, 000
1, 000, 000
100, 000, 000
∞
11. a little famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x→∞
x values
x
expression values
x ln(x)
1000
x ln(x)
100, 000
1, 000, 000
100, 000, 000
∞
12. subtle, easy, hard, & important Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim 5 x→∞
x values
x
expression values
5
1000
100, 000
1, 000, 000
100, 000, 000
∞
13. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. 5 x→∞ x lim
14. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
pg. 4
3x + 7 8−x
math hands
c
2007-2009 MathHands.com v.125
15. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
x−
p x2 + 8x − 5
16. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
x−
p x2 + 10x − 5
17. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
7 8 + 3x
18. not famous Find
sin(x) 5+ x→∞ x lim
19. not famous Find
sin(x) 5+ x→−∞ x lim
20. Find lim (12 sin(x)) x→∞
21. a little famous [ π2 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim tan−1 (x) x→∞
pg. 5
math hands
c
2007-2009 MathHands.com v.125
x values
x
expression values
tan−1 (x)
1000
100, 000
1, 000, 000
100, 000, 000
∞
22. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (.75)x − 1 x→∞ .75 − 1 lim
x values
x
expression values
(.75)x −1 .75−1
20
50
100
200
∞
23. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (.95)x − 1 x→∞ .95 − 1 lim
x values
x
expression values
(.95)x −1 .95−1
20
50
100
200
∞
24. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (1.05)x − 1 x→∞ 1.05 − 1 lim
pg. 6
x values
x
expression values
(1.05)x −1 1.05−1
20
math hands
50
100
200
∞
c
2007-2009 MathHands.com v.125
25. very famous rx − 1 x→∞ r − 1 lim
pg. 7
math hands
c
2007-2009 MathHands.com v.125
math hands Calculus: HW set 150c01s04 don’t try to find the answer, try to find meaning and understanding 1. Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. 5 x→∞ x lim
solution: x values
x
expression values
5 x
1000
100, 000
1, 000, 000
100, 000, 000
0.005
0.00005
0.000005
0.00000005
∞ by EM ≈ 0
Solution:
2. ridiculously famous [e] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x 1 lim 1 + x→∞ x
solution: x values expression values
x 1+
1 x x
1000
100, 000
1, 000, 000
100, 000, 000
2.71692393
2.71826824
2.71828047
2.71828181
∞ by EM ≈ 2.71828
Solution:
3. a little famous [e2 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit.
pg. 1
math hands
c
2007-2009 MathHands.com v.125
x 2 lim 1 + x→∞ x
solution: x values expression values
x 1+
2 x x
1000
100, 000
1, 000, 000
100, 000, 000
7.37431239
7.38890832
7.38904132
7.38905595
∞ by EM ≈ 7.38906
Solution:
4. a little famous [e−1 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x −1 1+ x→∞ x lim
solution: x values expression values
x 1+
−1 x x
1000
100, 000
1, 000, 000
100, 000, 000
0.36769542
0.3678776
0.36787926
0.36787944
∞ by EM ≈ 0.36788
Solution:
5. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. x1 1 lim x→∞ x
solution:
pg. 2
math hands
c
2007-2009 MathHands.com v.125
x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
0.99311605
0.99988488
0.99998618
0.99999982
x 1 x
x1
∞ by EM ≈ 1
Solution:
6. super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
1 x
solution: x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
0.999999833333
0.999999999983
1
1
x x sin
1 x
∞ by EM ≈ 1
Solution:
7. super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. 1 lim x 1 − cos x→∞ x
solution: x values expression values
pg. 3
x x 1 − cos
1 x
1000
100, 000
1, 000, 000
100, 000, 000
0.0000005
0.00000000005
0.000000000001
0
math hands
∞ by EM ≈ 0
c
2007-2009 MathHands.com v.125
Solution:
8. spoof on a super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
1 2x
solution: x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
0.499999979167
0.499999999998
0.500000000000
0.500000000000
x x sin
1 2x
∞
by EM ≈ 0.5
Solution:
9. spoof on a super ridiculously famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x sin x→∞
1 4x
solution: x values expression values
x x sin
1 4x
1000
100, 000
1, 000, 000
100, 000, 000
0.249999997396
0.250000000000
0.250000000000
0.250000000000
Solution:
pg. 4
math hands
c
2007-2009 MathHands.com v.125
∞
by EM ≈ 0.2
10. a little famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x→∞
ln(x) x
solution: x values
x
1000
100, 000
1, 000, 000
100, 000, 000
expression values
ln(x) x
0.0069078
0.0001151
0.0000138
0.0000002
∞ by EM ≈ 0
Solution:
11. a little famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim x→∞
x ln(x)
solution: x values
x
1000
100, 000
1, 000, 000
100, 000, 000
expression values
x ln(x)
144.7648273
8685.8896381
72382.4136505
5428681.0237906
∞ by EM ∞
Solution:
12. subtle, easy, hard, & important Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim 5 x→∞
solution:
pg. 5
math hands
c
2007-2009 MathHands.com v.125
x values
x
1000
100, 000
1, 000, 000
100, 000, 000
expression values
5
5.0000000
5.0000000
5.0000000
5.0000000
∞ by EM ≈ 5
Solution:
13. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. 5 x→∞ x lim
solution: PIM succeeds here. limx→∞
5 x
PIM
=
5 ∞
=0
Solution:
14. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
3x + 7 8−x
solution: PIM does NOT succeed here, since we get use EM for a sensible guess.
∞ −∞ ,
an indeterminate form, thus more work is needed. We
solution: x values expression values
x
3x+7 8−x
1000
100, 000
1, 000, 000
100, 000, 000
-3.03125
-3.00031002
-3.000031
-3.00000031
∞ by EM ≈ −3
Solution:
pg. 6
math hands
c
2007-2009 MathHands.com v.125
15. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
x−
p x2 + 8x − 5
solution:
√ PIM does NOT succeed here, since we get ∞ − ∞2 + 8∞ − 5 = ∞ − ∞, an indeterminate form, thus more work is needed. We use EM for a sensible guess. solution: x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
-3.989541778
-3.999895004
-3.9999895
-3.999999894
x x−
√ x2 + 8x − 5
∞ by EM ≈ −4
Solution:
16. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
x−
p x2 + 10x − 5
solution:
√ PIM does NOT succeed here, since we get ∞ − ∞2 + 10∞ − 5 = ∞ − ∞, an indeterminate form, thus more work is needed. We use EM for a sensible guess. solution: x values expression values
1000
100, 000
1, 000, 000
100, 000, 000
-4.985074516
-4.999850007
-4.999985
-4.999999849
x x−
√ x2 + 10x − 5
Solution:
17. Limits by PIM and EM combination: Attempt to determine the limit first by PIM, if PIM results in an indeterminate form, use EM to make a sensible guess as to the limit. lim x→∞
pg. 7
7 8 + 3x
math hands
c
2007-2009 MathHands.com v.125
∞ by EM ≈ −5
solution: PIM succeeds here. limx→∞
7 8+3x
PIM
=
7 8+3∞
=
7 8+∞
=
7 ∞
=0
Solution:
18. not famous Find
sin(x) 5+ x→∞ x lim
Solution: hint: graph it
Solution:
19. not famous Find
sin(x) lim 5+ x→−∞ x
Solution: hint: graph it
Solution:
20. Find lim (12 sin(x)) x→∞
Solution: hint: graph it
pg. 8
math hands
c
2007-2009 MathHands.com v.125
Solution:
21. a little famous [ π2 ] Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. lim tan−1 (x) x→∞
solution: x values
x
1000
100, 000
1, 000, 000
100, 000, 000
expression values
tan−1 (x)
1.56979633
1.57078633
1.57079533
1.57079632
∞ by EM ≈ 1.5708
Solution:
22. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (.75)x − 1 x→∞ .75 − 1 lim
solution: x values
x
20
50
100
200
expression values
(.75)x −1 .75−1
3.98731515
3.99999773
4
4
∞ by EM ≈ 4
Solution:
23. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit.
pg. 9
math hands
c
2007-2009 MathHands.com v.125
(.95)x − 1 x→∞ .95 − 1 lim
solution: x values
x
20
50
100
200
expression values
(.95)x −1 .95−1
12.83028155
18.46110049
19.88158942
19.99929895
∞ by EM ≈ 20
Solution:
24. very famous Limits by EM (engineer’s method ): Complete the table below for values of the expression. Based on that, make a sensible guess as to the indicated limit. (1.05)x − 1 x→∞ 1.05 − 1 lim
solution: x values
x
20
50
100
200
expression values
(1.05)x −1 1.05−1
33.0659541
209.34799572
2610.02515693
345831.6163032
∞ by EM ∞
Solution:
25. very famous rx − 1 x→∞ r − 1 lim
Solution: hint: your answer should begin such as ”depends on r, if r is ......??
pg. 10
math hands
c
2007-2009 MathHands.com v.125
Solution:
pg. 11
math hands
c
2007-2009 MathHands.com v.125
