1. Limits and Continuity

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^- 1 / 1 − x=∞

Evaluate lim x→0 x + 1/ 1 −cos x.

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.

g(2) =1,g(5) =−1,lim x→4 g(x) =−∞,lim x→7^− g(x) =∞,lim x→7^+ g(x) =−∞

{Use of Tech} Population growth Consider the following population functions.

d. Evaluate and interpret lim t→∞ p(t).

p(t) = 600 (t²+3/t²+9)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 3x) / x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 7x) / 3x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 5x) / x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂2 (sin (x-2)) / (x² - 4)

How is lim x🠂0 sin x/x used in this section?

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.

b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.

lim (4g(x))¹/³ = 2

x →0

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.

5 ―x²

lim ------------- = 0

x → ―2 (√g(x))