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# One-Sided 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.

Let f be a function and let a be a real number. Then the right-hand limit: f (x) represents the limit of f as x approaches a from the right, and the left-hand limit: f (x) represents the limit of f as x approaches a from the left.

### Existence of a Limit

Let f be a function and let a and L be real numbers. Then the two-sided limit: f (x ) = L if and only if the one-sided limits exist and f (x ) = f (x ) = L.

#### Example 1

Given find the limits: (a) f (x ), (b) f (x ), and (c) f (x ). Substituting x = 3 into f (x ) leads to a 0 in both the numerator and denominator. Factor which is equivalent to (x +1) where x ≠ 3. Thus, (a) f (x ) = (x +1) = 4, (b) f (x ) = (x +1) = 4, and (c) since the one-sided limits exist and are equal, f (x ) = f (x ) = 4, therefore the two-sided limit f (x ) exists and lim f (x ) = 4. (Note that f (x ) is undefined at x = 3, but the function gets arbitrarily close to 4 as x approaches 3. Therefore the limit exists.) (See Figure 5.1-3.)

#### Example 2

Given f (x ) as illustrated in the accompanying diagram (Figure 5.1-4), find the limits:

1. f (x ), (b) f (x), and (c) f (x).
1. As x approaches 0 from the left, f (x ) gets arbitrarily close to 0. Thus, f (x ) = 0.
2. As x approaches 0 from the right, f (x ) gets arbitrarily close to 2. Therefore, f (x ) = 2. Note that f (0) ≠ 2.
3. Since f (x ) ≠ f (x ), f (x ) does not exist.

#### Example 3

Given the greatest integer function f (x ) = [x], find the limits: (a) f (x ), (b) f (x ), and (c) f (x ).

1. Enter y1 = int(x) in your calculator. You see that as x approaches 1 from the right, the function stays at 1. Thus, [x] = 1. Note that f(1) is also equal 1.
2. As x approaches 1 from the left, the function stays at 0. Therefore, [x] = 0. Notice that [x] ≠ f (1).
3. Since [x] ≠ [x], therefore, [x] does not exist. (See Figure 5.1-5.)

#### Example 4

Given f (x) = , x ≠ 0, find the limits: (a) f (x ), (b) f (x ), and (c) f (x ).

#### Example 5

f (x ) = xe x = 0 and f (x ) = e 2x = 1. Thus, f (x ) does not exist.

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

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