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:

- Substituting directly
- Factoring and simplifying
- Multiplying the numerator and denominator of a rational function by the conjugate of either the numerator or denominator
- 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.

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