1. Limits and Continuity

Determine the following limits.

lim x→1^− x/ ln x

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.

_____

√x² + 4

c. g(x) = -----------

x

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→4 x^2 − 16 / 4 − x

Determine the following limits.

lim w→∞ (ln w²) / (ln w³ + 1)

Describe the end behavior of g(x) = e^-2x.

Determine the following limits. 

lim t→∞ (5t² + t sin t) / t²

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→4(3x−7)

Limit Evaluate lim x → ∞ (tanh x)ˣ.

Limits and Continuity

In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.

lim (x lim g(x)) = 2

x→-4 x→0

Evaluate each limit and justify your answer. 

lim x→0 (x^8−3x^6−1)^40

Analyze the following limits and find the vertical asymptotes of f(x) =(x − 5) / (x² − 25).

lim x→−5⁺ f(x)

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))

Determine the following limits.

a. lim z→3^+ (z − 1)(z − 2) / (z − 3)

6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

c. 1/√x

11. Show that if positive functions f(x) and g(x) grow at the same rate as x→∞, then f=O(g) and g=O(f).