Determine the following limits.​

lim x→∞ (x⁴ − 1) / (x⁵ + 2)

Evaluate each limit and justify your answer. 

lim x→5 ln 6(√x^2−16−3) / 5x−25

Estimate the following limits using graphs or tables.

lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

Determine the following limits.

$\underset{z\to 4}{\lim }\frac{z-5}{{(z^2-10z+24)}^2}$

Determine the following limits.

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

Determine the following limits at infinity.

lim x→−∞ 3x^11

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

Which of the following statements are correct? Choose all that apply.

a. lim x→1 1/ (x−1)^2 does not exist

b. lim x→1 1/ (x−1)^2=∞

c. lim x→1 1/(x−1)^2=−∞

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→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→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)

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)

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)

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

lim x→−2^− h(x)

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

lim x→−2^+ h(x)

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

lim x→−2 h(x)

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

lim x→3 h(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.

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