1. Limits and Continuity

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)

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.

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

Determine the following limits.

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

Determine the following limits.

a. $\underset{x\to 1^{+}}{\lim }\frac{x-3}{\sqrt{x^2-5x+4}}$

Determine the following limits.

b. $\underset{x\to 3}{\lim }\frac{x-3}{x^4-9x^2}$

Determine the following limits.

b. $\underset{x\to 2}{\lim }\frac{x-2}{x^5-4x^3}$

Determine the following limits.

$\underset{x\to 0}{\lim }\frac{x^3-5x^2}{x^2}$

Determine the following limits.

$\underset{x\to 1^{+}}{\lim }\frac{x^2-5x+6}{x-1}$

Determine the following limits.

$\underset{x\to 0^{-}}{\lim }\csc \left(x\right)$

Determine the following limits.

$\underset{\theta \to {\frac{\pi }{2}}^{+}}{\lim }\frac{1}{3}\tan \theta$

Determine the following limits.

$\underset{\theta \to 0}{\lim }\frac{2+\sin \theta }{1-{\cos }^2\theta }$

Evaluate each limit and justify your answer. 

lim x→0 (x^8−3x^6−1)^40

Evaluate each limit and justify your answer. 

lim x→2 (3 / 2x^5−4x^2−50)^4

Evaluate each limit and justify your answer. 

lim x→4 √x^3−2x^2−8x / x−4