1. Limits and Continuity

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

e. x^3 - x^2

82. Use the definitions of the hyperbolic functions to find each of the following limits.

a. lim(x→∞) tanh x

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?

a. log_2(x²)

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

lim x→9 √x − 3 / x − 9

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 f(x) / g(x)−h(x)

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

17. lim(x→∞)arcsec(x)

Determine the following limits.

lim x→1 x^3 − 7x^2 + 12x / 4 − x

Determine the following limits.

b. $\underset{x\to 2^{-}}{\lim }\frac{1}{\sqrt{x(x-2)}}$

Determine the following limits. 

lim x→−∞ (e^x cos x +3)

Which of the following is the correct algebraic method to find the limit $\underset{x\to 2}{lim}\frac{x^2-4}{x-2}$?

17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

Determine the following limits. 

lim x→∞ sin x / e^x

Find the exact length of the curve $y=\ln (\sec (x\left)\right)$ for $0\le x\le \frac{\pi }{4}$.

How many tangent lines to the curve $y=\frac{x}{x+1}$ pass through the point $(1,\frac{1}{2})$?