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# Infinite Limits for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

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

### Infinite Limits (as x → a)

If f is a function defined at every number in some open interval containing a, except possibly at a itself, then

1. means that f (x) increases without bound as x approaches a.
2. means that f (x) decreases without bound as x approaches a.

#### Limit Theorems

1. If n is a positive integer, then
2. If the , and then
3. If the and then
4. (Note that limit theorems 2 and 3 hold true for x → a + and x → a.)

#### Example 1

Evaluate the limit:

The limit of the numerator is 5 and the limit of the denominator is 0 through positive values. Thus, (b) The limit of the numerator is 5 and the limit of the denominator is 0 through negative values. Therefore, Verify your result with a calculator. (See Figure 5.2-1.)

#### Example 2

Find:

Factor the denominator obtaining The limit of the numerator is 9 and the limit of the denominator is (0)(6) = 0 through negative values. Therefore, Verify your result with a calculator. (See Figure 5.2-2.)

Find:

#### Example 4

Find: where [x] is the greatest integer value of x.

As x → 2, [x]=1. The limit of the numerator is (1 – 2) = –1. As x → 2, (2 – x)=0 through positive values. Thus,

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

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