1. Limits and Continuity

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

Evaluate each limit and justify your answer. 

lim t→4 t−4 /√t−2

Evaluate each limit and justify your answer. 

lim x→1 (x+5x / x+2)^4

Evaluate each limit and justify your answer. 

lim x→∞(2x+1x / x)^3

Evaluate each limit and justify your answer. 

lim x→5 ln 6(√x^2−16−3) / 5x−25

Evaluate each limit and justify your answer. 

lim x→0 (x / √16x+1-1)^1/3

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

a. lim x→−2^+ f(x)

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

b. lim x→−2 f(x)

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

d. lim x→0^+ f(x)

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

c. lim x→0^− f(x)

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