91Ó°ÊÓ

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

Find the oblique asymptote for $$ f(x)=\frac{3 x^{3}+4 x^{2}-x+1}{x^{2}+1} $$

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

Let \(R\) be the rectangle joining the midpoints of the sides of the quadrilateral \(Q\) having vertices \((\pm x, 0)\) and \((0, \pm 1)\). Calculate $$ \lim _{x \rightarrow 0^{+}} \frac{\text { perimeter of } R}{\text { perimeter of } Q} $$

Determine the largest interval over which the given function is continuous. $$ f(x)=\frac{1}{\sqrt{25-x^{2}}} $$

Using the symbols \(M\) and \(\delta\), give precise definitions of each expression. (a) \(\lim _{x \rightarrow c^{+}} f(x)=-\infty\) (b) \(\lim _{x \rightarrow c^{-}} f(x)=\infty\)

Verify that the given equations are identities. \(\sinh 2 x=2 \sinh x \cosh x\)

Let \(y=\sqrt{x}\) and consider the points \(M, N, O\), and \(P\) with coordinates \((1,0),(0,1),(0,0)\), and \((x, y)\) on the graph of \(y=\sqrt{x}\), respectively. Calculate (a) \(\lim _{x \rightarrow 0^{+}} \frac{\text { perimeter of } \Delta N O P}{\text { perimeter of } \Delta M O P}\) (b) \(\lim _{x \rightarrow 0^{+}} \frac{\text { area of } \Delta N O P}{\text { area of } \Delta M O P}\)

Determine the largest interval over which the given function is continuous. $$ f(x)=\sin ^{-1} x $$

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0}(\sin 5 x) / 3 x $$