1. Limits and Continuity

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

39. lim (x → ∞) (ln 2x - ln(x + 1))

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

a. limx→0+ (1 − cos x) / |cos x − 1|

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→3^+ h(x)

At what point do the curves $r_1\left(t\right)=(t,2-t,35+t^2)$ and $r_2\left(s\right)=(7-s,s-5,s^2)$ intersect?

Evaluate each limit. 

lim x→0 cos x−1 / sin^2x

Complete the following steps for the given functions. 

c. Graph $f$ and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f\left(x\right)=\frac{x^2-3}{x+6}$

Find the exact length of the curve $y=2\sqrt{3}{x}^{\frac{3}{2}}$ for $0\le x\le 4$.

Evaluate the following limit, if it exists. If the limit does not exist, select 'DNE'.
$$\underset{(x,y)\to (0,0)}{lim}\frac{xy}{x^2+y^2}$$

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

c. √(x^4 + x^3)

Evaluate each limit. 

$\underset{x\to 3}{\lim }\sqrt{x^2+7}$

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

95. lim(x→∞) (√(x² + x + 1) - √(x² - x))

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

limh→0− (√6 − √(5h² + 11h + 6))/ h

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

lim x→3 x − 3 /|x − 3|

Find the limit.

$\(\lim\)_{x\(\rarr\)5}\(\frac{x-5}{\sqrt{x}\)-\(\sqrt\)5}$

Evaluate each limit and justify your answer. 

lim x→2 (3 / 2x^5−4x^2−50)^4