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\(\lim\limits_{x\rightarrow 0}\) sin \(x =\) __________

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

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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 \(\lim\limits_{x\rightarrow c}g(x) =\) _________.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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\({\lim\limits_{x\rightarrow 0}} \dfrac{\sin \dfrac{{x}}{3}}{x}\)

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

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

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

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\(\lim\limits_{\theta \rightarrow 0}\dfrac{\tan \theta}{\theta}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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,

1. Find \(\lim\limits_{\theta \rightarrow {\pi}/{4}^{+}}x(\theta)\) and \(\lim\limits_{\theta \rightarrow {\pi}/{4} ^{+}}y (\theta)\).
2. Are the limits found in (a) consistent with what is expected physically?
3. Find \(\lim\limits_{\theta \rightarrow {\pi}/{2}^{-}}x(\theta)\) and \(\lim\limits_{\theta \rightarrow {\pi}/{2}^{-}}y(\theta)\).
4. Are the limits found in (c) consistent with what is expected physically?

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

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

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

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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 \(0\).

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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]\).

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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?

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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 \(0\)?

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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.)

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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 \(\lim\limits_{\theta \rightarrow 0^{-}}\sin \theta = 0.)\)

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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.

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

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

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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.

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

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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 \(\lim\limits_{x\rightarrow 0}[ xf(x)] =0\).

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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 \(0\)?