Infinite limits and limits at infinity AWS
INFINITE LIMITS AND LIMITS AT INFINITY Instructor: Mohammad Mashayekhi
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Infinite limits: ■ When the value of a function approaches to a large positive or negative number, the limit of the function is shown by +∞ or −∞ respectively. ■ Examples:
1 lim + = &→( 𝑥
lim. tan 𝜃
-→
+
+
sin 𝜃 + = lim. = cos 𝜃 -→ +
lim ln 𝑥 =
&→(7
Infinite one-sided limits: § Sometimes finding infinite one-sided limits are a little bit tricky! § Example: 𝑎) lim7
−5𝑥 + 1 𝑥 − 2
𝑏) lim>
−5𝑥 + 1 𝑥 − 2
&→+
&→+
Infinite limits: Some useful rules: § If lim 𝑓 𝑥 = +∞ , lim 𝑔 𝑥 = +∞ , lim ℎ 𝑥 = ℎ, and 𝑐 and ℎ are constant numbers: &→?
&→?
&→?
lim (𝑓 𝑥 + 𝑔 𝑥 ) = +∞ ,
&→?
lim (𝑓 𝑥 . 𝑔 𝑥 ) = +∞ ,
&→?
lim 𝑓 𝑥 − 𝑔 𝑥
&→?
K(&) &→? L(&)
lim
= 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒
+∞ 𝑐 > 0 lim 𝑐. 𝑓 𝑥 = N 0 𝑐 = 0 &→? −∞ 𝑐 < 0 lim
&→?
ℎ(𝑥) =0 𝑓(𝑥)
= 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒
Quiz! ■ What is the following limit: 𝑥+ + 𝑥 − 2 lim &→T+> (𝑥 + 2)U
a) +∞ b) −∞ c) 0 d) 𝐷𝑁𝐸
Quiz solution: § What is the following limit:
Correct answer: b) −∞
𝑥+ + 𝑥 − 2 lim &→T+> (𝑥 + 2)U
Quiz solution: § What is the following limit:
𝑥+ + 𝑥 − 2 lim &→T+> (𝑥 + 2)U
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Limits at infinity: § Sometimes we are interested to know behavior of a function for large positive or large negative 𝑥 values! § If the function approaches to a finite number for large x values either in positive or negative direction, the function is said to have horizontal asymptotes!
The most basic rules of limits at infinity:
lim 𝑐 = 𝑐
&→±\
,
] &→±\ &
lim
= 0
Quiz! ■ Which of the following statements is correct:
a)
] &→^\ & _
= 0 for all 𝑎
b)
] &→^\ & _
= 0 for all 𝑎 > 0
c)
] &→T\ & _
= 0 for all 𝑎
d)
] &→T\ & _
= 0 for all 𝑎 > 0
lim lim lim lim
Quiz solution: ■ Which of the following statements is correct: Correct answer: b)
a) b) c) d)
lim
]
lim
]
lim
]
lim
]
&→^\ & _ &→^\ & _ &→T\ & _ &→T\ & _
] &→^\ &_
lim
= 0 for all 𝑎 > 0
= 0 for all 𝑎 = 0 for all 𝑎 > 0 = 0 for all 𝑎 = 0 for all 𝑎 > 0
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Factoring technique: §
However sometimes the limit is indeterminate!
§
Examples of indeterminate expressions:
Strategy: factoring out the common factor usually solves the problem of indeterminate limits!
Factoring technique: §
` a
Example: lim 𝑥 − 𝑥
Ø Solution:
&→\
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Quotient functions involving polynomials: Let’s consider the following general form: 𝑃c (𝑥) 𝑎c 𝑥 c + 𝑎cT] 𝑥 cT] + ⋯ = 𝑃d (𝑥) 𝑎d 𝑥 d + 𝑎dT] 𝑥 dT] + ⋯ Using Factoring technique: 𝑃c (𝑥) = lim &→^\ 𝑃d (𝑥) &→^\ 𝑎d 𝑥 d lim
𝑎cT] 1 𝑎cT+ 1 + ⋯ 𝑎c 𝑥 𝑎c 𝑥 + = 𝑎dT] 1 𝑎dT+ 1 1+ + ⋯ 𝑎d 𝑥 𝑎d 𝑥 +
𝑎c 𝑥 c 1 +
There are three different results for this limit depending on 𝑛 and 𝑚
Quotient functions involving polynomials: a) 𝑛 > 𝑚 ?f &f lim &→^\ ?g &g
§ Example:
lim
+& iTj&^]
&→\ TU& k^U
?f . 𝑥 cTd &→^\ ?g
= lim
=
?f >0 ?g h ? −∞ 𝑖𝑓 f < 0 ?g
+∞ 𝑖𝑓
Quotient functions involving polynomials: b) 𝑛 < 𝑚 ?f &f lim &→^\ ?g &g
§ Example:
lim
Tj& k^U&
&→\ (+T&)(+& k^U&T])
?f ] . g>f = &→^\ ?g &
= lim
0
Quotient functions involving polynomials: c) 𝑛 = 𝑚 ?f &f lim &→^\ ?g &g
§ Example:
lim
T& k^U& iT]
&→\ Tl& iT]((((
=
?f ?g
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Example 1: § What is the following limit? j& k ^U&Tm lim UT& &→T\ Ø Solution:
Example 2: § Find limit of the following function for very large negative numbers:
𝑔 𝑧 = Ø Solution:
9𝑧 + − 15𝑧 + 3 + 3𝑧
Example 3: § Find the following limit:
lim
rst &
&→T\ +& u ^&T]
Ø Solution:
Example 4: § Find the following limit:
lim Ø Solution:
|]T+&| i
&→T\ (j i &^w)(T+ & k ^]( i &)
Example 5: § Find the following limit:
lim 𝑒 Ø Solution:
&→\
& k T+&T & k ^j
THANKS FOR YOUR TIME!
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Infinite limits: ■ When the value of a function approaches to a large positive or negative number, the limit of the function is shown by +∞ or −∞ respectively. ■ Examples:
1 lim + = &→( 𝑥
lim. tan 𝜃
-→
+
+
sin 𝜃 + = lim. = cos 𝜃 -→ +
lim ln 𝑥 =
&→(7
Infinite one-sided limits: § Sometimes finding infinite one-sided limits are a little bit tricky! § Example: 𝑎) lim7
−5𝑥 + 1 𝑥 − 2
𝑏) lim>
−5𝑥 + 1 𝑥 − 2
&→+
&→+
Infinite limits: Some useful rules: § If lim 𝑓 𝑥 = +∞ , lim 𝑔 𝑥 = +∞ , lim ℎ 𝑥 = ℎ, and 𝑐 and ℎ are constant numbers: &→?
&→?
&→?
lim (𝑓 𝑥 + 𝑔 𝑥 ) = +∞ ,
&→?
lim (𝑓 𝑥 . 𝑔 𝑥 ) = +∞ ,
&→?
lim 𝑓 𝑥 − 𝑔 𝑥
&→?
K(&) &→? L(&)
lim
= 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒
+∞ 𝑐 > 0 lim 𝑐. 𝑓 𝑥 = N 0 𝑐 = 0 &→? −∞ 𝑐 < 0 lim
&→?
ℎ(𝑥) =0 𝑓(𝑥)
= 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒
Quiz! ■ What is the following limit: 𝑥+ + 𝑥 − 2 lim &→T+> (𝑥 + 2)U
a) +∞ b) −∞ c) 0 d) 𝐷𝑁𝐸
Quiz solution: § What is the following limit:
Correct answer: b) −∞
𝑥+ + 𝑥 − 2 lim &→T+> (𝑥 + 2)U
Quiz solution: § What is the following limit:
𝑥+ + 𝑥 − 2 lim &→T+> (𝑥 + 2)U
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Limits at infinity: § Sometimes we are interested to know behavior of a function for large positive or large negative 𝑥 values! § If the function approaches to a finite number for large x values either in positive or negative direction, the function is said to have horizontal asymptotes!
The most basic rules of limits at infinity:
lim 𝑐 = 𝑐
&→±\
,
] &→±\ &
lim
= 0
Quiz! ■ Which of the following statements is correct:
a)
] &→^\ & _
= 0 for all 𝑎
b)
] &→^\ & _
= 0 for all 𝑎 > 0
c)
] &→T\ & _
= 0 for all 𝑎
d)
] &→T\ & _
= 0 for all 𝑎 > 0
lim lim lim lim
Quiz solution: ■ Which of the following statements is correct: Correct answer: b)
a) b) c) d)
lim
]
lim
]
lim
]
lim
]
&→^\ & _ &→^\ & _ &→T\ & _ &→T\ & _
] &→^\ &_
lim
= 0 for all 𝑎 > 0
= 0 for all 𝑎 = 0 for all 𝑎 > 0 = 0 for all 𝑎 = 0 for all 𝑎 > 0
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Factoring technique: §
However sometimes the limit is indeterminate!
§
Examples of indeterminate expressions:
Strategy: factoring out the common factor usually solves the problem of indeterminate limits!
Factoring technique: §
` a
Example: lim 𝑥 − 𝑥
Ø Solution:
&→\
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Quotient functions involving polynomials: Let’s consider the following general form: 𝑃c (𝑥) 𝑎c 𝑥 c + 𝑎cT] 𝑥 cT] + ⋯ = 𝑃d (𝑥) 𝑎d 𝑥 d + 𝑎dT] 𝑥 dT] + ⋯ Using Factoring technique: 𝑃c (𝑥) = lim &→^\ 𝑃d (𝑥) &→^\ 𝑎d 𝑥 d lim
𝑎cT] 1 𝑎cT+ 1 + ⋯ 𝑎c 𝑥 𝑎c 𝑥 + = 𝑎dT] 1 𝑎dT+ 1 1+ + ⋯ 𝑎d 𝑥 𝑎d 𝑥 +
𝑎c 𝑥 c 1 +
There are three different results for this limit depending on 𝑛 and 𝑚
Quotient functions involving polynomials: a) 𝑛 > 𝑚 ?f &f lim &→^\ ?g &g
§ Example:
lim
+& iTj&^]
&→\ TU& k^U
?f . 𝑥 cTd &→^\ ?g
= lim
=
?f >0 ?g h ? −∞ 𝑖𝑓 f < 0 ?g
+∞ 𝑖𝑓
Quotient functions involving polynomials: b) 𝑛 < 𝑚 ?f &f lim &→^\ ?g &g
§ Example:
lim
Tj& k^U&
&→\ (+T&)(+& k^U&T])
?f ] . g>f = &→^\ ?g &
= lim
0
Quotient functions involving polynomials: c) 𝑛 = 𝑚 ?f &f lim &→^\ ?g &g
§ Example:
lim
T& k^U& iT]
&→\ Tl& iT]((((
=
?f ?g
Outline: ■ Infinite limits ■ Limits at infinity ■ Factoring technique ■ Quotient functions involving polynomials ■ Examples
Example 1: § What is the following limit? j& k ^U&Tm lim UT& &→T\ Ø Solution:
Example 2: § Find limit of the following function for very large negative numbers:
𝑔 𝑧 = Ø Solution:
9𝑧 + − 15𝑧 + 3 + 3𝑧
Example 3: § Find the following limit:
lim
rst &
&→T\ +& u ^&T]
Ø Solution:
Example 4: § Find the following limit:
lim Ø Solution:
|]T+&| i
&→T\ (j i &^w)(T+ & k ^]( i &)
Example 5: § Find the following limit:
lim 𝑒 Ø Solution:
&→\
& k T+&T & k ^j
THANKS FOR YOUR TIME!
