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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 consider 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 the table below.
\(x\) \(-0.2\)
\(-0.1\)
\(-0.01\)
\(0.01\)
\(0.1\)
\(0.2\)
\(f(x)\)
(b) Write a sentence about what the table in
part a suggests regarding
\(\displaystyle\lim_{x\to 0}f(x)\text{.}\)
(c) Write a couple of sentences to explain how your values in the table in
part a and
\(\displaystyle\lim_{x\to 0}f(x)\) are demonstrated in the graph below.
(d) Let
\(\displaystyle g(x) = \frac{x}{|x|}\text{.}\) Fill in the blanks of the table below with the appropriate outputs of
\(g\) for values near
\(x = 0\text{.}\) Note that
\(g\) is not defined at
\(x=0\text{.}\)
\(x\) \(-0.1\)
\(-0.01\)
\(-0.001\)
\(0.001\)
\(0.01\)
\(0.1\)
\(g(x)\)
(e) Write a few sentences about what the values in the table in
part d suggests about
\(\displaystyle\lim_{x \to 0}g(x)\text{.}\)
(f)
Write a couple of sentences to explain how your values in the table in
part d and
\begin{equation*}
\lim_{x\to 0} g(x)
\end{equation*}
are demonstrated in the plot below.
(g) Let
\(\displaystyle h(x)=\frac{1-x^2}{x+1}\text{.}\) 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 parts.
