$\lim\limits_{x\rightarrow 0}$ sin $x =$ __________

True or False $\lim\limits_{x\rightarrow 0}\dfrac{\cos x-1}{x} = 1$.

The Squeeze Theorem states that if the functions $f,g$, and $h$ have the property $f(x)\leq g(x)\leq h(x)$ for all $x$ in an open interval containing $c$, except possibly at $c$, and if $\lim\limits_{x\rightarrow c}f(x) =$ $\lim\limits_{x\rightarrow c}h(x) = L$, then $\lim\limits_{x\rightarrow c}g(x) =$ _________.

True or False $f(x) = \csc x$ is continuous for all real numbers except $x = 0$.

Suppose $-x^{2}+1\leq g(x) \leq x^{2}+1$ for all $x$ in an open interval containing $0$. Find $\lim\limits_{x\rightarrow 0}g(x)$.

Suppose $-(x-2) ^{2}-3\leq g(x)\leq (x-2) ^{2}-3$ for all $x$ in an open interval containing $2$. Find $\lim\limits_{x\rightarrow 2}g(x)$.

Suppose $\cos x\leq g(x) \leq 1$ for all $x$ in an open interval containing $0$. Find $\lim\limits_{x\rightarrow 0}g(x)$.

Suppose $-x^{2}+1\leq g(x) \leq \sec x$ for all $x$ in an open interval containing $0$. Find $\lim\limits_{x\rightarrow 0}g(x)$.

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

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

${\lim\limits_{x\rightarrow \pi /3}}({\cos x+\sin x})$

${\lim\limits_{x\rightarrow \pi /3}}({\sin x-\cos x})$

${\lim\limits_{x\rightarrow 0}}\ \dfrac{\cos x}{1+\sin x}$

${\lim\limits_{x\rightarrow 0}}\ \dfrac{\sin x}{1+\cos x}$

$\lim\limits_{x\rightarrow 0}\dfrac{3}{1+e^{x}}$

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

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

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

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

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

$\lim\limits_{x\rightarrow 0}\dfrac{e^{2x}}{1+e^{x}}$

$\lim\limits_{x\rightarrow 0}\dfrac{1-e^{x}}{1-e^{2x}}$

${\lim\limits_{x\rightarrow 0}}\dfrac{\sin (7x)}{x}$

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

$\lim\limits_{\theta \rightarrow 0}\dfrac{\theta +3\sin \theta}{2\theta}$

${\lim\limits_{x\rightarrow 0}}\ \dfrac{2x-5\sin (3x)}{x}$

${\lim\limits_{\theta \rightarrow 0}}\ \dfrac{\sin \theta}{\theta +\tan \theta}$

$\lim\limits_{\theta \rightarrow 0}\dfrac{\tan \theta}{\theta}$

$\lim\limits_{\theta \rightarrow 0}\dfrac{5}{\theta \cdot \csc \theta}$

$\lim\limits_{\theta \rightarrow 0}\dfrac{\sin (3\theta)}{\sin (2\theta)}$

$\lim\limits_{\theta \rightarrow 0}\dfrac{1-\cos ^{2}\theta}{\theta}$

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

$\lim\limits_{\theta \rightarrow 0}(\theta \cdot \cot \theta)$

$\lim\limits_{\theta \rightarrow 0}\left[\sin \theta \left(\dfrac{\cot \theta -\csc \theta}{\theta}\right) \right]$

$f(x)=\left\{ \begin{array}{c@{\quad}l} 3\cos x & \hbox{ if } x\lt 0 \\[4pt] 3 & \hbox{ if }x=0 \\[4pt] x+3 & \hbox{ if }x>0 \end{array} \right. \quad at\quad c=0$

$f(x)=\left\{ \begin{array}{c@{\quad}l} \cos x & \hbox{ if }x\lt 0 \\[4pt] 0 & \hbox{ if }x=0 \\[4pt] e^{x} & \hbox{ if }x\gt 0 \end{array} \right. \quad at\quad c=0$

$f(\theta )=\left\{ \begin{array}{l@{\quad}l} \sin \theta & \hbox{ if }\theta \leq\dfrac{\pi }{4} \\[8pt] \cos \theta & \hbox{ if }\theta >\dfrac{\pi }{4} \end{array} \right. at\quad c=\dfrac{\pi }{4}$

$f(x)=\left\{ \begin{array}{c@{\quad}l} \tan ^{-1}x & \hbox{ if }x\lt 1 \\[4pt] \ln x & \hbox{ if }x\geq 1 \end{array} \right. at\quad c=1$

$f(x) = \sin \left(\dfrac{x^{2}-4x}{x-4}\right)$

$f(x) = \cos \left(\dfrac{x^{2}-5x+1}{2x}\right)$

$f(\theta) = \dfrac{1}{1+\sin \theta}$

$f(\theta) = \dfrac{1}{1+\cos^{2}\theta}$

$f(x) = \dfrac{\ln x}{x-3}$

$f(x) = \ln (x^{2}+1)$

$f(x) = e^{-x}\sin x$

$f(x) = \dfrac{e^{x}}{1+\sin ^{2}x}$

$\lim\limits_{x\rightarrow 0}\left(x^{2}\sin \dfrac{1}{x}\right)$

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

$\lim\limits_{x\rightarrow 0}\left[ x^{2}\left(1-\cos \dfrac{1}{x}\right) \right]$

$\lim\limits_{x\rightarrow 0}\left[ \sqrt{x^{3}+3x^{2}}\sin \left(\dfrac{1}{x}\right) \right]$

$\lim\limits_{x\rightarrow 0}\dfrac{\sin (ax)}{\sin (bx)} = \dfrac{a}{b}$;  $b\neq0$

$\lim\limits_{x\rightarrow 0}\dfrac{\cos (ax)}{\cos (bx)} = 1\qquad$

$\lim\limits_{x\rightarrow 0}\dfrac{\sin (ax)}{bx} = \dfrac{a}{b}$;  $b\neq 0$

$\lim\limits_{x\rightarrow 0}\dfrac{1-\cos (ax)}{bx} = 0$ $a\neq 0$, $b\neq 0$

Projectile Motion An object is propelled from ground level at an angle $\theta$, $\dfrac{\pi}{4}\lt \theta \lt \dfrac{\pi}{2}$, up a ramp that is inclined to the horizontal at an angle of $45^{\circ}$. See the figure. If the object has an initial velocity of 10 feet/second, the equations of the horizontal position $x = x(\theta)$ and the vertical position $y = y(\theta)$ of the object after $t$ seconds are given by $$\begin{equation*} x = x(\theta) = (10\cos \theta) t \quad \hbox{and} \quad y = y(\theta) = -16t^{2}+ (10\sin \theta) t \end{equation*}$$

For $t$ fixed,

Show that $\lim\limits_{x\rightarrow 0}\dfrac{1-\cos x}{ x^{2}} = \dfrac{1}{2}$.

Squeeze Theorem If $0\leq f(x)\leq 1$ for every number $x$, show that $\lim\limits_{x\rightarrow 0}[ x^{2}f(x)] = 0$.

Squeeze Theorem If $0\leq f(x)\leq M$ for every $x$, show that $\lim\limits_{x\rightarrow 0}[ x^{2}f(x)] = 0$.

The function $f(x) = \dfrac{\sin (\pi x)}{x}$ is not defined at $0$. Decide how to define $f(0)$ so that $f$ is continuous at $0$.

Define $f(0)$ and $f(1)$ so that the function $f(x) = \dfrac{\sin (\pi x)}{x(1-x)}$ is continuous on the interval $[0,1]\).

Is $f(x)=\left\{ \begin{array}{c@{\quad}l} \dfrac{\sin x}{x} & \hbox{if }x\neq 0 \\[10pt] 1 & \hbox{if }x=0 \end{array} \right.$ continuous at 0?

Is $f(x) = {\left\{ \begin{array}{c@{\quad}l} \dfrac{1-\cos x}{x} & \hbox{if }x\neq 0 \\[10pt] 0 & \hbox{if }x = 0 \end{array} \right.}$ continuous at $0$?

Squeeze Theorem Show that $\lim\limits_{x\rightarrow0}\left[x^{n}\sin \left(\dfrac{1}{x}\right) \right] = 0$, where $n$ is a positive integer. (Hint: Look first at Problem 57.)

Prove $\lim\limits_{\theta \rightarrow 0}\sin \theta = 0.$ (Hint: Use a unit circle as shown in the figure, first assuming $0\lt \theta \,\lt \dfrac{\pi}{2}$. Then use the fact that $\sin \theta$ is less than the length of the arc $AP$, and the Squeeze Theorem, to show that $\lim\limits_{\theta \rightarrow 0^{+}}\sin \theta = 0$. Then use a similar argument with $-\dfrac{\pi}{2}\lt \theta \lt 0$ to show $\lim\limits_{\theta \rightarrow 0^{-}}\sin \theta = 0.)$

Prove $\lim\limits_{\theta \rightarrow 0}\cos \theta = 1$. Use either the proof outlined in Problem 64 or a proof using the result $\lim\limits_{\theta \rightarrow 0}\sin \theta = 0$ and a Pythagorean identity.

Without using limits, explain how you can decide whether $f(x) = \cos (5x^{3}+2x^{2}-8x+1)$ is continuous.

Explain the Squeeze Theorem. Draw a graph to illustrate your explanation.

Use the Sum Formulas $\sin (a+b) = \sin a\cos b+\cos a\sin b$ and $\cos (a+b) = \cos a\cos b-\sin a\sin b$ to show that the sine function and cosine function are continuous on their domains.

Find $\lim\limits_{x\rightarrow 0}\dfrac{\sin x^{2}}{x}$.

Squeeze Theorem If $f(x)=\left\{ \begin{array}{l@{\quad}l} 1 & \hbox{if }x\hbox{ is rational} \\[3pt] 0 & \hbox{if }x\hbox{ is irrational} \end{array} \right.$ show that $\lim\limits_{x\rightarrow 0}[ xf(x)] =0$.

Suppose points $A$ and $B$ with coordinates $(0,0)$ and $(1,0)$, respectively, are given. Let $n$ be a number greater than $0$, and let $\theta$ be an angle with the property $0\lt \theta \lt \dfrac{\pi}{1+n}$. Construct a triangle $ABC$ where $\overline{AC}$ and $\overline{AB}$ form the angle $\theta$, and $\overline{CB}$ and $\overline{AB}$ form the angle $n\,\theta$ (see the figure below). Let $D$ be the point of intersection of $\overline{AB}$ with the perpendicular from $C$ to $\overline{AB}$. What is the limiting position of $D$ as $\theta$ approaches $0$?