1. Limits and Continuity

Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find

b. limx→c 2f(x)g(x)

Calculating Limits

Find the limits in Exercises 11–22.

limt→6 8(t−5)(t−7)

Calculating Limits

Find the limits in Exercises 11–22.

limx→−1/2 4x(3x+4)²

Limits of quotients

Find the limits in Exercises 23–42.

limx→−5 (x² + 3x − 10) / x + 5

Limits of quotients

Find the limits in Exercises 23–42.

limx→−1 (√(x² + 8) − 3) / (x + 1)

Limits of quotients

Find the limits in Exercises 23–42.

limx→−3 (2 − √(x² − 5)) / (x + 3)

Limits with trigonometric functions

Find the limits in Exercises 43–50.

lim x→0 tan x

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→−π √(x + 4) cos(x + π)

Suppose limx→b f(x) = 7 and lim x→b g(x) = −3. Find

b. limx→b f(x)⋅g(x)

Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find

a. limx→−2 (p(x) + r(x) + s(x))

Theory and Examples

If limx→4 (f(x) − 5) / (x − 2) = 1, find limx→4 f(x).

Theory and Examples

If limx→−2 f(x) / x² = 1, find

b. limx→−2 f(x) / x

Theory and Examples

a. If limx→0 f(x) / x² = 1, find limx→0 f(x).

Suppose limx→4 f(x) = 0 and lim x→4 g(x) = −3. Find

c. limx→4 (g(x))²

Using Limit Rules

Suppose lim x→0 f(x) = 1 and lim x→0 g(x) = −5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.

limx→0 (2f(x) − g(x)) / (f(x) + 7)² = limx→0 (2f(x) − g(x)) / limx→0 (f(x) + 7)² (a)

(We assume the denominator is nonzero.)

(lim x→0 2f(x) − lim x→0 g(x)) / (lim x→0 (f(x) + 7))² (b)

= (2 lim x→0 f(x) − lim x→0 g(x)) / (lim x→0 f(x) + lim x→0 7)² (c)

= ((2)(1) − (−5)) / (1 + 7)² = 7/64