91Ó°ÊÓ

. Let $$ f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x \text { is rational } \\ x^{4} & \text { if } x \text { is irrational } \end{array}\right. $$ For what values of \(a\) does \(\lim _{x \rightarrow a} f(x)\) exist?

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

Verify that the given equations are identities. \(e^{x}=\cosh x+\sinh x\)

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

Verify that the given equations are identities. \(e^{2 x}=\cosh 2 x+\sinh 2 x\)

, 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 non-removable. $$ f(x)=\frac{x^{2}-100}{x-10} ; c=10 $$

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.

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

Find each of the following limits or state that it does not exist. (a) \(\lim _{x \rightarrow 1} \frac{|x-1|}{x-1}\) (b) \(\lim _{x \rightarrow 1^{-}} \frac{|x-1|}{x-1}\) (c) \(\lim _{x \rightarrow 1^{-}} \frac{x^{2}-|x-1|-1}{|x-1|}\) (d) \(\lim _{x \rightarrow 1^{-}}\left[\frac{1}{x-1}-\frac{1}{|x-1|}\right]\)