1. Limits and Continuity

Determine the following limits.

a. lim x→4^+ x − 5 / (x − 4)^2

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (1 + x + sin x) / (3 cosx)

Determine the following limits.

$\underset{t\to \infty }{\lim }\frac{\cos \left(t\right)}{e^{3t}}$

Determine the following limits.

lim p→1 p^5 − 1 / p − 1

Evaluate each limit and justify your answer. 

lim t→4 t−4 /√t−2

13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.

1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

e. (3/2)^x

Find constants b and c in the polynomial p(x)=x^2+bx+c such that lim x→2 p(x) / x−2=6. Are the constants unique?

Determine the following limits. 

lim x→−∞ (3x⁷ + x²) 

Determine the following limits.

b. lim x→4^− x − 5 / (x − 4)^2

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 ((4―g(x)) / x ) = 1

x→0

Determine the following limits.

lim x→π/2 1/√sin x − 1 / x + π/2

Evaluate each limit. 

lim x→−1 (x^2−4+ 3√x^2−9)

Behavior at the origin Using calculus and accurate sketches, explain how the graphs of f(x) = xᵖ ln x differ as x → 0⁺ for p = 1/2, 1, and 2.

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x²/2 − 1/x) as

d. x→−1