Evaluate lim x→∞ f(x) and lim x→−∞ f(x) sing the figure. <IMAGE>

Evaluate each limit. 

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

Suppose x lies in the interval (1, 3) with x≠2. Find the smallest positive value of δ such that the inequality 0<|x−2|<δ is true.

Evaluate $\underset{x\to \infty }{\lim }f\left(x\right)$ and$\underset{x\to -\infty }{\lim }f\left(x\right)$.

$f\left(x\right)=\frac{4x^3+1}{1-x^3}$

Evaluate each limit and justify your answer. 

lim x→0 (x^8−3x^6−1)^40

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

$\underset{x\to 0}{\lim }\frac{1-\cos \left(x\right)}{{\cos }^2\left(x\right)-3\cos \left(x\right)+2}$

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→3 x^3=27