91Ó°ÊÓ

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow-3^{+}} \frac{\sqrt{3+x}}{x}$$

Find the limits. \(\lim _{x \rightarrow 0^{-}} \frac{1+\cos x}{\sin x}\)

In Problems 41-52, verify that the given equations are identities. e^{2 x}=\cosh 2 x+\sinh 2 x$

The function \(f(x)=x^{2}\) had been carefully graphed, but during the night a mysterious visitor changed the values of \(f\) at a million different places. Does this affect the value of \(\lim _{x \rightarrow a} f(x)\) at any \(a\) ? Explain.

Find the limits. \(\lim _{x \rightarrow \infty} \frac{\sin x}{x}\)

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ f(x)=\frac{x^{2}-100}{x-10} ; c=10 $$

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow-\pi^{+}} \frac{\sqrt{\pi^{3}+x^{3}}}{x}$$

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ f(x)=\frac{\sin x}{x} ; c=0 $$

In Problems 41-52, verify that the given equations are identities. \(e^{-x}=\cosh x-\sinh x\)

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(f(x)=\frac{3}{x+1}\)