1. Limits and Continuity

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

lim x→1 5x^2 + 6x + 1 / 8x − 4

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1^− f(x)

Determine the following limits.

lim x→∞ (2x − 3) / (4x + 10)

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

c. x²e^(-x)

Evaluate lim x→0 x + 1/ 1 −cos x.

Evaluate each limit. 

$\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(x\right)-1}{\sqrt{\sin \left(x\right)}-1}$

Determine the following limits.

lim x→3^- x − 4 / x^2 − 3x

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

c. lim(x→∞) sinh 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 1/x−1/4 / x − 4

Determine the following limits.

lim x→−5^+ x − 5 / x + 5

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))

Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find

b. limx→c 2f(x)g(x)

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1^+ f(x)

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→2^+ f(x)

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^- 1 / 1 − x=∞