2.4: The Precise Definition of a Limit
- Page ID
- 143310
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- Suppose a function \(g(x)\) is continuous at \(x = b\), with \(g(b) = M\). Write what this means in terms of the \(\varepsilon\)-\(\delta\) definition of continuity.
- Suppose \(\displaystyle \lim_{x \to b} g(x) = M\) for some function \(g(x)\). Write what this means in terms of the \(\varepsilon\)-\(\delta\) definition of the limit.
- Suppose a function \(H(\phi)\) is continuous at \(\phi =\pi\), with \(H(\pi) = P\). Write what this means in terms of the \(\varepsilon\)-\(\delta\) definition of continuity.
- Suppose \(\displaystyle \lim_{\phi \to \pi} H(\phi) = P\) for some function \(H(\phi)\). Write what this means in terms of the \(\varepsilon\)-\(\delta\) definition of the limit.
- Using a graphing calculator, find a number \(\delta > 0\) so that \(|x^2 - 4| < 0.5\) whenever \(|x - 2| < \delta\).
- Using a graphing calculator, find a number \(\delta > 0\) so that \(\left|\sin(x) - \dfrac{1}{2}\right| < 0.1\) whenever \(\left|x - \dfrac{\pi}{6}\right| < \delta\).
- Consider the function \(f(x) = \dfrac{x^3 - x}{|x|}\).
- Using a graphing calculator, find a number \(\delta > 0\) so that \(\left|f(x) - 1\right| < 3\) whenever \(0 < |x - 0| < \delta\).
- Does your result from (a) mean that \(\displaystyle \lim_{x \to 0} f(x) = 1\)? Explain.
- Is it possible to find a number \(\delta > 0\) so that \(\left|f(x) - 1\right| < 1\) whenever \( 0 < |x - 0| < \delta\)? Explain.
- Using a graphing calculator, find a number \(\delta > 0\) so that \(\left|f(x) - 1\right| < 3\) whenever \(0 < |x - 0| < \delta\).
- Using the \(\varepsilon\)-\(\delta\) definition of the limit, prove that \(\displaystyle \lim_{x \to -3} (2x - 1) = -7\).
- Using the \(\varepsilon\)-\(\delta\) definition of the limit, prove that \(\displaystyle \lim_{x \to -4} \dfrac{x^2 - 16}{x + 4} = -8\).
- Using the \(\varepsilon\)-\(\delta\) definition of the limit, prove that \(\displaystyle \lim_{x \to 2} \dfrac{x^2 - 3x + 2}{x - 2} = 1\).
- Using the \(\varepsilon\)-\(\delta\) definition of the limit, prove that \(\displaystyle \lim_{x \to -1} x^2 = 1\).
- Using the \(\varepsilon\)-\(\delta\) definition of the limit, prove that \(\displaystyle \lim_{x \to 1} (x^2 + 3x - 2) = 2\).