91Ó°ÊÓ

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. $$ f(x)=\frac{4-x}{2-\sqrt{x}} ; c=4 $$

Verify that the given equations are identities. \(\cosh (x-y)=\cosh x \cosh y-\sinh x \sinh y\)

, find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow 3^{+}}\left[x^{2}+2 x\right] $$

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. $$ g(x)=\frac{2 x}{\sqrt{x^{2}+5}} $$

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0^{+}} x^{x} $$

In Problems \(49-54\), determine the largest interval over which the given function is continuous. $$ f(x)=\sqrt{25-x^{2}} $$

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} \sqrt{|x|} $$

Suppose that \(f(x) g(x)=1\) for all \(x\) and \(\lim _{x \rightarrow a} g(x)=0\). Prove that \(\lim _{x \rightarrow a} f(x)\) does not exist.

The line \(y=a x+b\) is called an oblique asymptote to the graph of \(y=f(x)\) if either \(\lim _{x \rightarrow \infty}[f(x)-(a x+b)]=0\) or \(\lim _{x \rightarrow-\infty}[f(x)-(a x+b)]=0 .\) Find the oblique asymptote for $$ f(x)=\frac{2 x^{4}+3 x^{3}-2 x-4}{x^{3}-1} $$

Verify that the given equations are identities. \(\tanh (x+y)=\frac{\tanh x+\tanh y}{1+\tanh x \tanh y}\)