1. Limits and Continuity

Determine the following limits.

b. lim x→3^− 2/(x − 3)^3

Suppose g(x) = {2x+1 if x≠0

5 if x=0.

Compute g(0) and lim x→0 g(x)

Evaluate each limit and justify your answer. 

lim x→4 √x^3−2x^2−8x / x−4

Determine the following limits.

lim u→0^+ u − 1 / sin u

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?

c. √x

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).

lim x → -7 f(x) 

7. Order the following functions from slowest growing to fastest growing as x→∞.

a. e^x

b. x^x

c. (ln x)^x

d. e^(x/2)

112. True, or false? Give reasons for your answers.

a. 1/x⁴ = O(1/x² + 1/x⁴)

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))

Evaluate the limit, if it exists: $\underset{(x,y)\to (0,0)}{lim}\frac{xy}{x^2+y^2}$.

Evaluate each limit and justify your answer. 

lim x→∞(2x+1x / x)^3

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

lim x→b (x − b)^50 − x + b / x − b

Given the function $f\left(x\right)=2x^2-3x+1$, which of the following is the average rate of change of $f\left(x\right)$ over the interval from $x=-6$ to $x=-3$?

Find the exact length of the curve $y=x-x^2+{\sin -1}^x$ from $x=0$ to $x=1$.