True or False $\dfrac{f(x) }{g(x) }$ is an indeterminate form at $c$ of the type $\dfrac{0}{0}$ if $\lim\limits_{x\rightarrow c}\dfrac{f(x) }{g(x) }$ does not exist.

True or False If $\dfrac{f(x) }{g(x) }$ is an indeterminate form at $c$ of the type $\dfrac{0}{0},$ then L’Hôpital’s Rule states that $\lim\limits_{x\rightarrow c}\dfrac{f(x) }{g(x) }= \lim\limits_{x\rightarrow c}\left[ \dfrac{d}{dx}\left( \dfrac{f(x) }{g(x) }\right) \right]$.

True or False $\dfrac{1}{x}$ is an indeterminate form at $0$.

True or False $x\ln x$ is not an indeterminate form at $0^{+}$ because $\lim\limits_{x\rightarrow 0^{+}}x=0$ and $\lim\limits_{x\rightarrow 0^{+}}\ln x=-\infty$, and $0\cdot -\infty =0$.

In your own words, explain why $\infty -\infty$ is an indeterminate form, but $\infty +\infty$ is not an indeterminate form.

In your own words, explain why $0\cdot \infty \neq 0.$

$\dfrac{1-e^{x}}{x},$ $c=0$

$\dfrac{1-e^{x}}{x-1}$, $c=0$

$\dfrac{e^{x}}{x}$, $c=0$

$\dfrac{e^{x}}{x},$ $c=\infty$

$\dfrac{\ln x}{x^{2}}$, $c=\infty$

$\dfrac{\ln ( x+1) }{e^{x}-1},$ $c=0$

$\dfrac{\sec x}{x},$$c=0$

$\dfrac{x}{\sec x-1},$ $c=0$

$\dfrac{\sin x( 1-\cos x) }{x^{2}},$ $c=0$

$\dfrac{\sin x-1}{\cos x}$, $c=\dfrac{\pi }{2}$

$\dfrac{\tan x-1}{\sin ( 4x-\pi ) },$ $c=\dfrac{\pi }{4}$

$\dfrac{e^{x}-e^{-x}}{1-\cos x},$$c=0$

$x^{2}e^{-x},$ $c=\infty$

$x\;\cot\;x,$ $c=0$

$\csc \dfrac{x}{2}-\cot \dfrac{x}{2},$ $c=0$

$\dfrac{x}{x-1}+\dfrac{1}{\ln x},$ $c=1$

$\left( \dfrac{1}{x^{2}}\right) ^{\sin x},$ $c=0$

$(e^{x}+x)^{1/x},$ $c=0$

$( x^{2}-1) ^{x},$ $c=0$

$( \sin x) ^{x},$ $c=0$

$\lim\limits_{x\rightarrow 2}\dfrac{x^{2}+x-6}{x^{2}-3x+2}$

$\lim\limits_{x\rightarrow 1}\dfrac{2x^{3}+5x^{2}-4x-3}{x^{3}+x^{2}-10x+8}$

$\lim\limits_{x\rightarrow 1}\dfrac{\ln x}{x^{2}-1}$

$\lim\limits_{x\rightarrow 0}\dfrac{\ln (1-x)}{e^{x}-1}$

$\lim\limits_{x\rightarrow 0}\dfrac{e^{x}-e^{-x}}{\sin x}$

$\lim\limits_{x\rightarrow 0}\dfrac{\tan (2x) }{\ln (1+x)}$

$\lim\limits_{x\rightarrow 1}\dfrac{\sin ( \pi x) }{x-1}$

$\lim\limits_{x\rightarrow \pi }\dfrac{1+\cos x}{\sin (2x) }$

$\lim\limits_{x\rightarrow \infty }\dfrac{x^{2}}{e^{x}}$

$\lim\limits_{x\rightarrow \infty } \dfrac{e^{x}}{x^{4}}$

$\lim\limits_{x\rightarrow \infty }\dfrac{\ln x}{e^{x}}$

$\lim\limits_{x\rightarrow \infty }\dfrac{x+\ln x}{x\ln x}$

$\lim\limits_{x\rightarrow 0}\dfrac{e^{x}-1-\!\sin x}{1-\cos x}$

$\lim\limits_{x\rightarrow 0}\dfrac{e^{x}-e^{-x}-2\;\sin\;x}{3x^{3}}$

$\lim\limits_{x\rightarrow 0}\dfrac{\sin x-x}{x^{3}}$

$\lim\limits_{x\rightarrow 0}\dfrac{x^{3}}{\cos x-1}$

$\lim\limits_{x\rightarrow 0^{+}}(x^{2}\;\ln\;x)$

$\lim\limits_{x\rightarrow \infty }(xe^{-x})$

$\lim\limits_{x\rightarrow \infty} [x(e^{1/x}-1)]$

$\lim\limits_{x\rightarrow \pi /2} [ (1-\!\sin x)\tan x]$

$\lim\limits_{x\rightarrow \pi /2}(\sec x-\tan x)$

$\lim\limits_{x\rightarrow 0}\left( \cot x-\dfrac{1}{x}\right)$

$\lim\limits_{x\rightarrow 1}\left( \dfrac{1}{\ln x}-\dfrac{x}{\ln x}\right)$

$\lim\limits_{x\rightarrow 0}\left( \dfrac{1}{x}-\dfrac{1}{e^{x}-1}\right)$

$\lim\limits_{x\rightarrow 0^{+}}(2x)^{3x}$

$\lim\limits_{x\rightarrow 0^{+}}x^{x^{2}}$

$\lim\limits_{x\rightarrow \infty }( x+1)^{e^{-x}}$

$\lim\limits_{x\rightarrow \infty }(1+x^{2})^{1/x}$

$\lim\limits_{x\rightarrow 0^{+}}(\csc x)^{\sin x}$

$\lim\limits_{x\rightarrow \infty }x^{1/x}$

$\lim\limits_{x\rightarrow \pi /2^{-}}(\sin x)^{\tan x}$

$\lim\limits_{x\rightarrow 0}(\cos x)^{1/x}$

$\lim\limits_{x\rightarrow 0^{+}}\dfrac{\cot x}{\cot (2x) }$

$\lim\limits_{x\rightarrow \,\infty }\dfrac{\ln (\ln x)}{\ln x}$

$\lim\limits_{x\rightarrow 1/2^{-}}\dfrac{\ln (1-2x)}{\tan ( \pi x) }$

$\lim\limits_{x\rightarrow 1^{-}}\dfrac{\ln (1-x)}{\cot ( \pi x) }$

$\lim\limits_{x\rightarrow \infty }\dfrac{x^{4}+x^{3}}{e^{x}+1}$

$\lim\limits_{x\rightarrow \infty }\dfrac{x^{2}+x-1}{e^{x}+ e^{-x}}$

$\lim\limits_{x\rightarrow 0}\dfrac{xe^{4x}-x}{1-\cos (2x) }$

$\lim\limits_{x\rightarrow 0}\dfrac{x\tan x}{1-\cos x}$

$\lim\limits_{x\rightarrow 0}\dfrac{\tan ^{-1}x}{x}$

$\lim\limits_{x\rightarrow 0}\dfrac{\tan ^{-1}x}{\sin ^{-1}x}$

$\lim\limits_{x\rightarrow 0}\dfrac{\cos x-1}{\cos (2x) -1}$

$\lim\limits_{x\rightarrow 0}\dfrac{\tan x-\!\sin x}{x^{3}}$

$\lim\limits_{x\rightarrow 0^{+}}(x^{1/2} \ \ln x)$

$\lim\limits_{x\rightarrow \infty } [ (x-1)e^{-x^{2}}]$

$\lim\limits_{x\rightarrow \pi /2} [ \tan x \ \ln (\sin x) ]$

$\lim\limits_{x\rightarrow 0^{+}}[ \sin x \ \ln (\sin x)]$

$\lim\limits_{x\rightarrow 0}\ [ \csc x \ \ln (x+1)]$

$\lim\limits_{x\rightarrow \pi /4}[(1-\tan x) \ \sec (2x) ]$

$\lim\limits_{x\rightarrow a}\left[ (a^{2}-x^{2}) \ \tan \left( \dfrac{\pi x}{2a}\right) \right]$

$\lim\limits_{x\rightarrow 1^{+}}\,\left[ (1-x)\tan \!\left(\!\dfrac{1}{2}\pi x\!\right)\! \right]$

$\lim\limits_{x\rightarrow 1}\left( \dfrac{1}{\ln x}-\dfrac{1}{x-1}\right)$

$\lim\limits_{x\rightarrow 1}\left(\! \dfrac{x}{x-1}-\dfrac{1}{\ln x}\! \right)$

$\lim\limits_{x\rightarrow \pi /2}\left( x\;\tan\;x-\dfrac{\pi }{2}\sec x\right)$

$\lim\limits_{x\rightarrow \pi }(\cot\;x-x\;\csc\;x)$

$\lim\limits_{x\rightarrow 1^{-}}(1-x)^{\tan\;( \pi x) }$

$\lim\limits_{x\rightarrow 0^{+}}x^{\sqrt{\scriptstyle x}}$

$\lim\limits_{x\rightarrow 0}\left( \dfrac{\sin\;x}{x}\right) ^{\!\!1/x}$

$\lim\limits_{x\rightarrow \infty }\left(\! 1+\dfrac{5}{x}+\dfrac{3}{x^{2}}\!\right) ^{\!\!x}$

$\lim\limits_{x\rightarrow ( \pi /2) ^{-}}( \tan x) ^{\cos x}$

$\lim\limits_{x\rightarrow 0^{+}}( x^{2}+x) ^{-\ln x}$

$\lim\limits_{x\rightarrow 0}( \cosh x) ^{e^{x}}$

$\lim\limits_{x\rightarrow 0^{+}}(\sinh x) ^{x}$

Wolf Population In 2002 there were $65$ wolves in Wyoming outside of Yellowstone National Park, and in 2010 there were $247$ wolves. Suppose the population $w$ of wolves in the region at time $t$ follows the logistic growth curve $$w=w(t)=\dfrac{Ke^{rt}}{\dfrac{K}{40}+e^{rt}-1}$$

where $K=366$, $r=0.283$, and $t=0$ represents the population in the year 2000.

Skydiving The downward velocity $v$ of a skydiver with nonlinear air resistance can be modeled by $$v=v(t) =-A+RA\frac{e^{Bt+C}-1}{e^{Bt+C}+1}$$

where $t$ is the time in seconds, and $A,B,C,$ and $R$ are positive constants with $R>1$.

Electricity  The equation governing the amount of current $I$ (in amperes) in a simple RL circuit consisting of a resistance $R$ (in ohms), an inductance $L$ (in henrys), and an electromotive force $E$ (in volts) is $\ I=\dfrac{E}{R}( 1-e^{-Rt/L})$.

Find $\lim\limits_{x\rightarrow 0}\dfrac{a^{x}-b^{x}}{x},$ where $a\neq 1$ and $b\neq 1$ are positive real numbers.

Show that $\lim\limits_{x\,\rightarrow \,\infty } \frac{\ln x}{x^{n}}=0$, for $n \ge 1$ an integer.

Show that $\lim\limits_{x\rightarrow \,\infty } \frac{x^{n}}{e^{x}}=0$ for $n \ge 1$ an integer.

Show that $\lim\limits_{x\rightarrow 0^{+}}(\cos\;x+2\;\sin x)^{\cot\;x}=e^{2}$.

Find $\lim\limits_{x\rightarrow \,\infty } \dfrac{P(x)}{e^{x}}$, where $P$ is a polynomial function.

Find $\lim\limits_{x\rightarrow \infty }\left[ \ln (x+1)-\ln (x-1)\right] .$

Show that $\lim\limits_{x\rightarrow 0^{+}}\dfrac{e^{-1/x^{2}}}{x}=0$. Hint: Write $\dfrac{e^{-1/x^{2}}}{x}=\dfrac{\dfrac{1}{x}}{e^{1/x^{2}}}.$

If $n$ is an integer, show that $\lim\limits_{x\rightarrow 0^{+}}\dfrac{e^{-1/x^{2}}}{x^{n}}=0$.

Show that $\lim\limits_{x\rightarrow \infty }\sqrt[x]{x}=1.$

Show that $\lim\limits_{x\rightarrow \infty }\left( 1+\dfrac{a}{x}\right) ^{\!\!x}=e^{a},$ $a$ any real number.

Show that $\lim\limits_{x\rightarrow \infty }\left( \dfrac{x+a}{x-a}\right) ^{\!\!x}= e^{2a}, a\ne 0$.

If $a,b\neq 0$ and $c>0$ are real numbers, show that $$\lim\limits_{x\rightarrow c}\dfrac{x^{a}-c^{a}}{x^{b}-c^{b}}=\dfrac{a}{b} c^{a-b}.$$

Prove L’Hôpital’s rule when $\dfrac{f(x)}{g(x) }$ is an indeterminate form at $-\infty$ of the type $\dfrac{0}{0}$.

Explain why L’Hôpital’s Rule does not apply to $\lim\limits_{x\rightarrow 0}\dfrac{x^{2}\sin \dfrac{1}{x}}{\sin x}.$

Find each limit:

Find constants $A,$ $B$, $C$, and $D$ so that $$\lim\limits_{x\rightarrow 0}\dfrac{\sin ( Ax) +Bx+Cx^{2}+Dx^{3}}{x^{5}}=\dfrac{4}{15}.$$

A function $f$ has derivatives of all orders.

The formulas in Problem 111 can be used to approximate derivatives. Approximate $f^\prime (2),$ $\ f^{\prime \prime} (2)$, and $f’’’ (2)$ from the table. The data are for $f(x)=\ln x$. Compare the exact values with your approximations.

Consider the function $f( t,x) =\dfrac{x^{t+1}-1}{t+1},$ where $x>0$ and $t\neq -1.$