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Limits at Infinity for AP Calculus

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.

If f is a function defined at every number in some interval (a,∞), then f (x)= L means that L is the limit of f (x) as x increases without bound.

If f is a function defined at every number in some interval (– ∞, a), then f (x)= L means that L is the limit of f (x) as x decreases without bound.

Limit Theorem

If n is a positive integer, then

Example 1

Evaluate the limit:

Divide every term in the numerator and denominator by the highest power of x (in this case, it is x ), and obtain:

Verify your result with a calculator. (See Figure 5.2-3.)

Limits at Infinity

Example 2

Evaluate the limit:

Divide every term in the numerator and denominator by the highest power of x. In this

Verify your result with a calculator. (See Figure 5.2-4.)

Limits at Infinity

Example 3

Evaluate the limit:

Divide every term in the numerator and denominator by the highest power of x. In this

of the numerator is –1 and the limit of the denominator is 0. Thus,

Verify your result with a calculator. (See Figure 5.2-5.)

Limits at Infinity

Example 4

Evaluate the limit:

As x → –∞, x < 0 and thus, x = – Divide the numerator and denominator by x (not x2 since the denominator has a square root). Thus, you

Verify your result with a calculator. (See Figure 5.2-6.)

Limits at Infinity

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

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