Chapter 2

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

. 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. $$ \lim _{x \rightarrow-3^{+}} \frac{\sqrt{3+x}}{x} $$

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

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

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

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 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. $$ f(x)=\frac{\sin x}{x} ; c=0 $$

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. $$ f(x)=\frac{3}{x+1} $$

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) \(\lim _{x \rightarrow 1^{-}}\left[\frac{1}{x-1}-\frac{1}{|x-1|}\right]\)

, find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow 3^{+}} \frac{x-3}{\sqrt{x^{2}-9}} $$

Determine the largest interval over which the given function is continuous. $$ f(x)=\frac{\cos x}{x} ; c=0 $$

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. $$ f(x)=\frac{\cos x}{x} ; c=0 $$

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

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. $$ f(x)=\frac{3}{(x+1)^{2}} $$

, find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow 1^{-}} \frac{\sqrt{1+x}}{4+4 x} $$

Determine the largest interval over which the given function is continuous. $$ g(x)=\left\\{\begin{array}{ll} \frac{\sin x}{x}, & x \neq 0 \\ 0, & x=0 \end{array}\right. $$

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

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. $$ F(x)=\frac{2 x}{x-3} $$

Determine the largest interval over which the given function is continuous. $$ F(x)=x \sin \frac{1}{x} ; c=0 $$

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. $$ F(x)=x \sin \frac{1}{x} ; c=0 $$

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

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. $$ F(x)=\frac{3}{9-x^{2}} $$

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

Determine the largest interval over which the given function is continuous. $$ f(x)=\sin \frac{1}{x} ; c=0 $$

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

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. $$ g(x)=\frac{14}{2 x^{2}+7} $$

, find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow 0^{-}} \frac{x}{|x|} $$

Determine whether the function is continuous at the given point \(c .\) If the function is not continuous, determine whether the discontinuity is removable or 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 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 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 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. $$ f(x)=\sqrt{25-x^{2}} $$

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

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 $$ f(x)=\frac{2 x^{4}+3 x^{3}-2 x-4}{x^{3}-1} $$

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.

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

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

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 $$ f(x)=\frac{3 x^{3}+4 x^{2}-x+1}{x^{2}+1} $$

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 $$ \lim _{x \rightarrow 0^{+}} \frac{\text { perimeter of } R}{\text { perimeter of } Q} $$

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

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

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

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

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. $$ f(x)=\operatorname{sech} x $$