2.2: The Limit of a Function
- Page ID
- 143298
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Sketch the graph of a function \(f(x)\) that is continuous at \(x = 1\) with \(\displaystyle \lim_{x \to 1} f(x) = -2\).
- Sketch the graph of a function \(g(x)\) that is discontinuous at \(x = 1\) with \(\displaystyle \lim_{x \to 1} g(x) = -2\).
- Sketch the graph of a function \(h(x)\) with \(\displaystyle \lim_{x \to -2^-} h(x) = 3\), \(\displaystyle \lim_{x \to -2^+} h(x) = 1\), and \(h(-2) = 3\).
- Sketch the graph of a function \(f(t)\) with \(\displaystyle \lim_{t \to 0} f(t) = -\infty\).
- Sketch the graph of a function \(g(t)\) with \(\displaystyle \lim_{t \to -1^-} g(t) = \infty\), \(\displaystyle \lim_{t \to -1^+} g(t) = 0\), and \(g(1) = 2\).
- Given only that \(\displaystyle \lim_{x \to 11} h(x) = 4\) for some function \(h(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11^-} h(x)\) and \(\displaystyle \lim_{x \to 11^+} h(x)\)? Briefly explain why or why not.
- Given only that \(\displaystyle \lim_{x \to 11} f(x) = 4\) for some function \(f(x)\), is it possible to calculate \(f(11)\)? Briefly explain why or why not.
- Given only that \(g(11) = 4\) for some function \(g(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11} g(x)\)? Briefly explain why or why not.
- Given only that \(h(11) = 4\) for some continuous function \(h(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11} h(x)\)? Briefly explain why or why not.
- Given only that \(\displaystyle \lim_{x \to 11^-} f(x) = 4\) and \(\displaystyle \lim_{x \to 11^+} f(x) = 4\) for some function \(f(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11} f(x)\)? Briefly explain why or why not.
- Given only that \(\displaystyle \lim_{x \to 11^-} f(x) = 5\) and \(\displaystyle \lim_{x \to 11^+} f(x) = 2\) for some function \(f(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11} f(x)\)? Briefly explain why or why not.
- Use a calculator to complete the table below for the function \(g(x) = 2x + 1\), then use your results to make inferences about \(\displaystyle \lim_{x \to -1^-} g(x)\), \(\displaystyle \lim_{x \to -1^+} g(x)\), and \(\displaystyle \lim_{x \to -1} g(x)\).
\(x\) \(g(x)\) \(x\) \(g(x)\) -1.01 -0.99 -1.001 -0.999 -1.0001 -0.9999
- Use a calculator to complete the table below for the function \(h(x) = \dfrac{\sin(x)}{x}\), then use your results to make inferences about \(\displaystyle \lim_{x \to 0^-} h(x)\), \(\displaystyle \lim_{x \to 0^+} h(x)\), and \(\displaystyle \lim_{x \to 0} h(x)\).
\(x\) \(h(x)\) \(x\) \(h(x)\) -0.01 0.01 -0.001 0.001 -0.0001 0.0001
- Use a calculator to complete the table below for the function \(f(x) = \dfrac{x^2-1)}{|x-1|}\), then use your results to make inferences about \(\displaystyle \lim_{x \to 1^-} f(x)\), \(\displaystyle \lim_{x \to 1^+} f(x)\), and \(\displaystyle \lim_{x \to 1} f(x)\).
\(x\) \(f(x)\) \(x\) \(f(x)\) 0.99 1.01 0.999 1.001 0.9999 1.0001
- Use a calculator to complete the table below for the function \(g(x) = (1+x)^{1/x}\), then use your results to make inferences about \(\displaystyle \lim_{x \to 0^-} g(x)\), \(\displaystyle \lim_{x \to 0^+} g(x)\), and \(\displaystyle \lim_{x \to 0} g(x)\).
\(x\) \(g(x)\) \(x\) \(g(x)\) -0.01 0.01 -0.001 0.001 -0.0001 0.0001
- Use a calculator to complete the table below for the function \(h(x) = \dfrac{x+1}{x-2}\), then use your results to make inferences about \(\displaystyle \lim_{x \to 2^-} h(x)\), \(\displaystyle \lim_{x \to 2^+} h(x)\), and \(\displaystyle \lim_{x \to 2} h(x)\).
\(x\) \(h(x)\) \(x\) \(h(x)\) 1.99 2.01 1.999 2.001 1.9999 2.0001