limits of functions

Calculus Limits Lesson 2

Bell Activity A. Use your calculator graph to find:

1. limx2 (3x  2)

2. limx3 (2 x  5)

x 3. lim x  4 ( 4  ) 2 2 ( x  4) 4. lim x 2 ( x  2)

B. Without Calculator, Find 1.

f(2) if f(x) = 3x – 2

2.

f(-3) if f(x) = 2x + 5

3.

f(4) if f(x) = 4 – x/2

4.

2 ( x  4) f(2) if f(x) = ( x  2)

Notice: In #4, both the numerator and denominator are 0. This is called an indeterminate form and thus cannot be evaluated.

Which of these can be substituted directly to find the limit? 1.

2.

3.

lim x 3 ( x 2  2) =11 x2 1 lim x 2 x 1 2 x 1 lim x 1 x 1

=3 =2

Even though # 3’s limit can’t be found by substituting directly we can still get the limit from a graph or a table.

Perhaps we could modify the problem from #3 before we substituted.

x 1 x 1 2

lim x 1

( x  1)( x  1) lim x1 x 1

lim x1 ( x  1)

=2

Modify these and find the limit by substituting:

x  x6 2 x 9

lim x3

x 1 x 1 3

2

lim x1

Let’s verify these with the calculator graph.

Find the limit:

x 1 x 1 2

lim x  1

Let’s verify with the calculator graph.

There are other times when we cannot find the limit by either substituting or factoring.

limx0 In this case, Rationalize the Numerator

x  1 1 x

Find the limit:

Let’s verify with the calculator graph.

And, of course, there are problems which merely need to be simplified before substituting.

( x  x)  x x 3

limx0

3

Assignment 

Finish Problems (a) through (m) in packet