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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Preview Activity 11.2.1 .
In this Preview Activity, we will relook at several ideas related to limits of (single variable) functions by working with tables, graphs, and the algebraic forms of these functions.
(a) Let
\(f\) defined by
\(f(x) = 3-x\text{.}\) Complete
Table 11.2.1 .
Table 11.2.1. Values of \(f(x) = 3-x\text{.}\)
\(x\) \(-0.2\)
\(-0.1\)
\(-0.01\)
\(0.01\)
\(0.1\)
\(0.2\)
\(f(x)\)
(b)
\begin{equation*}
\lim_{x\to 0}f(x)
\end{equation*}
(c)
Write a couple of sentences to explain how your values in
Table 11.2.1 and the
\begin{equation*}
\lim_{x\to 0}f(x)
\end{equation*}
Figure 11.2.4. A plot of \(f(x)=3-x\)
(d)
Let
\begin{equation*}
g(x) = \frac{x}{|x|}\text{.}
\end{equation*}
Fill in the blanks of
Table 11.2.5 with the appropriate outputs of
\(g\) for values near
\(x = 0\text{.}\) Note that
\(g\) is not defined at
\(x=0\text{.}\)
Table 11.2.5. Values of \(g(x) = \frac{x}{|x|}\text{.}\)
\(x\) \(-0.1\)
\(-0.01\)
\(-0.001\)
\(0.001\)
\(0.01\)
\(0.1\)
\(g(x)\)
(e) Write a few sentences about what the values in
Table 11.2.5 suggests about
\(\lim_{x \to 0}g(x)\text{.}\)
(f)
Write a couple of sentences to explain how your values in
Table 11.2.5 and the
\begin{equation*}
\lim_{x\to 0} g(x)
\end{equation*}
Figure 11.2.8. A plot of \(g(x)=\frac{x}{|x|}\)
(g) Let
\(h(x)=\frac{1-x^2}{x+1}\text{.}\) At what point(s) is
\(h\) undefined?
(h) Show that the limit as
\(x\) goes to
\(-1\) of
\(h(x)\) exists. Write a sentence or two about why
\(h(-1)\) does not exist but the limit as
\(x\) goes to
\(-1\) does exist.
(i) Graph
\(h(x)\) for a region around
\(x=-1\) and explain how this graph relates to your answer to the previous two tasks.
