AP Calculus Techniques for Limits

Limits (Part 1)

Basic Calculus

Objectives

• Evaluate limits using the table of values

• Illustrate the limit of a function using its graph

• Distinguish between   $\lim_{x\rightarrow c}\ f\left(x\right)$  and  $f\left(c\right)$  using the table of value

• Distinguish between   $\lim_{x\rightarrow c}\ f\left(x\right)$  and  $f\left(c\right)$  using the graph of  $y=f\left(x\right)$  

Limits

The idea of limits is fundamental to the study of calculus. It suggests a boundary that may be reached or not, or possibly exceeded. A mathematical limit has characteristics similar to those of a physical limit. It is the analysis of how function values or outputs change, when inputs change.

Limits

In symbols, we write this process as
 $\lim_{x\rightarrow c}\ f\left(x\right)=L$  
This is read, ‘‘The limit of f(x) as x approaches c is L.”

Example

Evaluate the  $\lim_{x\rightarrow2}\left(1+3x\right)$  using the table of values

Imagine a number line, Substitute any number close to 2 from the left

Let's say x=1.9, then  $\lim_{x\rightarrow2}\left(1+3x\right)=6.7$  
We get the another number closer than the first still from the left of 2

So, we get x=1.99, then  $\lim_{x\rightarrow2}\left(1+3x\right)=6.97$  
We continue doing so until the fourth time to see the pattern

Multiple Choice

What number is the $$\lim_{x\rightarrow2}\left(1+3x\right)$$  approaches from the left?

• 6

• 6.9

• 7

• 7.1

Multiple Choice

What number is the  $$\lim_{x\rightarrow2}\left(1+3x\right)$$  approaches from the right?

• 7

• 7.3

• 7.6

• 8

Multiple Choice

Given that  $$f\left(x\right)=1+3x$$  , what is  $$f\left(2\right)$$  ?

• 6

• 7

• 8

Limits

The limits of the function exists if
(i) the  $\lim_{x\rightarrow c^{\ -\ }}f\left(x\right)$  exists

(ii) the  $\lim_{x\rightarrow c\ ^{+\ }}f\left(x\right)$  exists
(iii)  $\lim_{x\rightarrow c^{\ -\ }}f\left(x\right)$  =  $\lim_{x\rightarrow c\ ^{+\ }}f\left(x\right)$  

Multiple Choice

What is the limit of  $$\lim_{x\rightarrow-1}\left(x^2+1\right)$$  ?

• 1

• 2

• 3

• 4