Evaluating Limits for AP Calculus

Practice problems for this concept can be found at Limits and Continuity Practice Problems for AP Calculus.

If f is a continuous function on an open interval containing the number a, then .

Common techniques in evaluating limits are:

1. Substituting directly
2. Factoring and simplifying
3. Multiplying the numerator and denominator of a rational function by the conjugate of either the numerator or denominator
4. Using a graph or a table of values of the given function.

Example 1

Find the limit: .

Substituting directly: .

Example 2

Find the limit: .

Using the product rule, .

Example 3

Find the limit: .

Factoring and simplifying: .

(Note that had you substituted t = 2 directly in the original expression, you would have obtained a zero in both the numerator and denominator.)

Example 4

Find the limit: .

Factoring and simplifying: .

Example 5

Find the limit: .

Multiplying both the numerator and the denominator by the conjugate of the numerator, .

(Note that substituting 0 directly into the original expression would have produced a 0 in both the numerator and denominator.)

Example 6

Find the limit: .

Enter in the calculator. You see that the graph of f (x) approaches 3 as x approaches 0. Thus, the . (Note that had you substituted x = 0 directly in the original expression, you would have obtained a zero in both the numerator and denominator.) (See Figure 5.1-1.)

Example 7

Find the limit: .

Enter into your calculator. You notice that as x approaches 3 from the right, the graph of f (x) goes higher and higher, and that as x approaches 3 from the left, the graph of f (x) goes lower and lower. Therefore, is undefined. (See Figure 5.1-2.)

Practice problems for this concept can be found at Limits and Continuity Practice Problems for AP Calculus.