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Use a table of numbers to investigate \({\lim\limits_{x \rightarrow 0}}\dfrac{1-\cos x}{1+ \cos x}\).

Question

\(f(x)=\left\{ \begin{array}{l@{\quad}l@{\quad}l} 2x-5 & & \hbox{if }x\lt1 \\ 6-9x & & \hbox{if }x\geq 1 \end{array} \right.\) \(\quad\) at \(c=1\)

Question

\(f(x)=\left\{ \begin{array}{l@{\quad}l@{\quad}l} x^{2}+2 & & \hbox{if }x\lt2 \\ 2x+1 & & \hbox{if }x\geq 2 \end{array} \right.\) \(\quad\) at \(c=2\)

Question

For \(f(x)=x^{2}-3\):

1. Find the slope of the secant line joining \((1,-2)\) and \((2,1)\).
2. Find the slope of the tangent line to the graph of \(f\) at \((1,-2)\).

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

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\(f(x)=3x^{2}+2x\)

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Find \(\lim\limits_{x\rightarrow 0}f(x)\) if \( 1+\sin x\leq f(x)\leq \vert x\vert +1\)

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\(\lim\limits_{x\rightarrow 2}\) \(\left( 2x-\dfrac{1}{x}\right)\)

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\(\lim\limits_{x\rightarrow \pi } \left( x\cos x\right)\)

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\(\lim\limits_{x\rightarrow -1}\left( x^{3}+3x^{2}-x-1\right)\)

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\(\lim\limits_{x\rightarrow 0}\sqrt[3]{x( x+2) ^{3}}\)

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\({\lim\limits_{x\rightarrow 0}}\) \([(2x+3) (x^{5}+5x)]\)

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\({\lim\limits_{x\rightarrow 3}} \dfrac{x^{3}-27}{x-3}\)

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\({\lim\limits_{x\rightarrow 3}}\left(\dfrac{x^{2}}{x-3}-\dfrac{3x}{x-3}\right)\)

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\({\lim\limits_{x\rightarrow 2}}\dfrac{x^{2}-4}{x-2}\)

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\({\lim\limits_{x\rightarrow -1}}\dfrac{x^{2}+3x+2} {x^{2}+4x+3}\)

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\({\lim\limits_{x\rightarrow -2}} \dfrac{x^{3}+5x^{2}+6x}{x^{2}+x-2}\)

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\({\lim\limits_{x\rightarrow 1}} \left(x^{2}-3x+\dfrac{1}{x}\right) ^{15}\)

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\({\lim\limits_{x\rightarrow 2}}\dfrac{3-\sqrt{x^{2}+5}}{x^{2}-4}\)

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\({\lim\limits_{x\rightarrow 0}} \left\{ \dfrac{1}{x}\left[ \dfrac{1}{(2+x)^{2}}-\dfrac{1}{4}\right] \right\}\)

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\({\lim\limits_{x\rightarrow 0}} \dfrac{\left({x+3}\right) ^{2}-9}{x}\)

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\({\lim\limits_{x\rightarrow 1}} [(x^{3}-3x^{2}+3x-1) (x+1) ^{2}]\)

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\({\lim\limits_{x\rightarrow - 2^{+}}}\dfrac{ x^{2}+5x+6}{x+2}\)

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\({\lim\limits_{x \rightarrow 5^{+}}}\dfrac{|x-5|}{x-5}\)

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\({\lim\limits_{x\rightarrow 1^{-}}}\dfrac{|x-1|}{x-1}\)

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\({\lim\limits_{x\rightarrow \,3/2^{+}}}\lfloor 2x\rfloor\)

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\({\lim\limits_{x\rightarrow 4^{-}}}\dfrac{x^{2}-16}{x-4}\)

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\({\lim\limits_{x\rightarrow 1^{+}}},\sqrt{x-1}\)

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\(f(x)=\left\{ \begin{array}{l@{\quad}l} 2x+3 & \hbox{if }x\lt 2 \\[3pt] 9-x & \hbox{if }x\geq 2 \end{array} \right.\) at \(c\)=2

Question

\(f(x)=\left\{ \begin{array}{c@{ }c} 3x+1 & \hbox{if }x \lt 3 \\[3pt] 10 & \hbox{if }x=3 \\[3pt] 4x-2 & \hbox{if }x \gt 3 \end{array} \right.\) at \(c=3\)

Question

\(f(x)=\left\{ \begin{array}{c@{}c} 5x-2 & \hbox{if }x \lt 1 \\[3pt] 5 & \hbox{if }x=1 \\[3pt] 2x+1 & \hbox{if }x \gt 1 \end{array} \right.\) at \(c=1\)

Question

\(f(x)=\left\{ \begin{array}{c@{}c} x^{2} & \hbox{if }x \lt -1 \\[3pt] 2 & \hbox{if }x=-1 \\[3pt] -3x-2 & \hbox{if }x \gt -1 \end{array} \right.\) at \(c=-1\)

Question

\(f(x)=\left\{ \begin{array}{c@{}l} 4-3x^{2} & \hbox{if }x\lt0 \\[3pt] 4 & \hbox{if }x=0 \\[3pt] \sqrt{16-x^{2}} & \hbox{if }0 \lt x\leq 4 \end{array} \right.\) at \(c=0\)

Question

\(f(x)=\left\{ \begin{array}{c@{}l} \sqrt{4+x} & \hbox{if }-4\leq x\leq 4 \\[3pt] \sqrt{\dfrac{x^{2}-16}{x-4}} & \hbox{if }x>4 \end{array} \right.\) at \(c=4\)

Question

\(f(x) =\) \(\lfloor \,2x\rfloor\) at \(c=\dfrac{1}{ 2}\)

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\(f(x)=|\,x-5\,|\) at \(c=5\)

Question

1. Find the average rate of change of \(f(x)= 2x^2 -5x\) from 1 to \(x\).
2. Find the limit as \(x\) approaches 1 of the average rate of change found in (a).

Question

A function \(f\) is defined on the interval \([-1,1]\) with the following properties: \(f\) is continuous on \([-1,1]\) except at \(0\), negative at \(-1\), positive at \(1\), but with no zeros. Does this contradict the Intermediate Value theorem?

Question

\(f(x)=\dfrac{x}{x^{3}-27}\)

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\(f(x)=\dfrac{x^{2}-3}{x^{2}+5x+6}\)

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\(f(x)=\dfrac{2x+1}{x^{3}+4x^{2}+4x}\)

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\(f(x) =\sqrt{x-1}\)

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\(f(x) =2^{-x}\)

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Use the Intermediate Value Theorem to determine whether \( 2x^{3}+3x^{2}-23x-42=0\) has a zero in the interval \([3,4]\).

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\(f(x) =8x^{4}-2x^{2}+5x-1\) on the interval \(\left[0,1\right]\).

Question

\(f(x) =3x^{3}-10x+9;\) zero between \(-3\) and \(-2\).

Question

Find \({\lim\limits_{x\rightarrow 0^{+}}}\dfrac{|\,x\,|}{x}(1-x)\) and \({\lim\limits_{x\rightarrow 0^{-}}}\dfrac{|\,x\,|}{x}(1-x)\). What can you say about \({\lim\limits_{x\rightarrow 0}}\dfrac{|x|}{x} (1-x)\)?

Question

Find \({\lim\limits_{x\rightarrow 2}}\left( \dfrac{x^{2}}{x-2}- \dfrac{2x}{x-2}\right)\). Then comment on the statement that this limit is given by \(\lim\limits_{x\rightarrow 2}\dfrac{x^{2}}{x-2}-\lim\limits_{x\rightarrow 2}\dfrac{2x}{x-2}\).

Question

Find \({\lim\limits_{h\rightarrow 0}} \dfrac{f(x+h)-f(x)}{h}\) for \(f(x)=\sqrt{x}\).

Question

For \(\lim\limits_{x\rightarrow 3}(2x+1)=7\), find the largest possible \(\delta\) that “works” for \(\epsilon =0.01\).

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\(\lim\limits_{x\rightarrow 0}\cos\) (tan x)

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\(\lim\limits_{x\rightarrow 0}{\dfrac{{\sin {\dfrac{{x}}{{4}}}}}{{x}}}\)

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\(\lim\limits_{x\rightarrow 0}\,\dfrac{\tan (3x) }{\tan ( 4x) }\)

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\({\lim\limits_{x\rightarrow 0}}\dfrac{\cos {\dfrac{x}{3}-1}}{x}\)

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\(\lim\limits_{x\rightarrow 0}\left( \dfrac{\cos x-1}{x}\right) ^{10}\)

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\(\lim\limits_{x\rightarrow 0}{\dfrac{{e^{4x}-1}}{e^{x}{-1}}}\)

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\(\lim\limits_{x\rightarrow \pi /2^{+}}\tan x\)

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\(\lim\limits_{x\rightarrow -3}\dfrac{2+x}{( x+3) ^{2}}\)

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\(\lim\limits_{x\rightarrow \infty }\dfrac{3x^{3}-2x+1}{x^{3}-8}\)

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\(\lim\limits_{x\rightarrow \infty }\dfrac{3x^{4}+x}{2x^{2}}\)

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\(f(x)=\dfrac{4x-2}{x+3}\)

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

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Let \(f( x) =\left\{ \begin{array}{c@{}c} \dfrac{\tan x}{2x} & \hbox{if }x\neq 0 \\ \dfrac{1}{2} & \hbox{if }x=0 \end{array} \right.\). Is \(f\) continuous at \(0\)?

Question

Let \(f( x) =\left\{ \begin{array}{c@{}c} \dfrac{\sin ( 3x) }{x} & \hbox{if }x\neq 0 \\ 1 & \hbox{if }x=0 \end{array} \right.\). Is \(f\) continuous at \(0\)?

Question

The function \(f( x) =\dfrac{\cos \left( \pi x+\dfrac{\pi }{2}\right) }{x}\) is not defined at \(0\). Decide how to define \(f( 0)\) so that \(f\) is continuous at \(0\).

Question

Use an \(\epsilon\) - \(\delta\) argument to show that the statement \(\lim\limits_{x\rightarrow -3} (x^{2}-9) =-18\) is false.

Question

1. Sketch a graph of a function \(f\) that has the following properties: \begin{eqnarray*} &&\\[-30pt] &&f(-1)=0,\quad \lim_{x\rightarrow \infty }f(x)=2,\quad \lim_{x\rightarrow -\infty }f(x)=2,\\[5pt] &&\lim_{x\rightarrow 4^{-}}f(x)=-\infty, \quad \lim_{x\rightarrow 4^{+}}f(x)=\infty \end{eqnarray*}

Question

1. Find the domain and the intercepts (if any) of \(R( x) =\dfrac{2x^{2}-5x+2}{5x^{2}-x-2}\).
2. Discuss the behavior of the graph of \(R\) at numbers where \(R\) is not defined.
3. Find any vertical or horizontal asymptotes of the function \(R\).

Question

If \(1-x^{2}\leq f(x)\leq\) cos \(x\) for all \(x\) in the interval \(-\dfrac{\pi }{2} \lt x \lt \dfrac{\pi }{2}\), show that \(\lim\limits_{x\rightarrow 0}f(x)=1\).