The Limit of a Function

Read this section for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4.

Practice Problem Answers

Practice 1: (a) 2  (b) 2  (c) \text{does not exist} (no limit)  (d) 1


Practice 2:

(a) \lim\limits_{x \rightarrow 2} \frac{(x+1)(x-2)}{x-2}=\lim\limits_{x \rightarrow 2}(x+1)=3

(b) \lim\limits_{t \rightarrow 0} \frac{t \cdot \sin (t)}{t(t+3)}=\lim\limits_{t \rightarrow 0} \frac{\sin (t)}{t+3}=\frac{0}{3}=0

(c) \lim\limits_{w \rightarrow 2} \frac{w-2}{\ln (w / 2)}=2. Try this one numerically or using a graph.

\begin{array}{l|l} {\mathrm{w}} & \frac{\mathrm{w}-2}{\ln (\mathrm{w} / 2)} \\ \hline 2.2 & 2.098411737 \\ 2.01 & 2.004995844 \\ 2.003 & 2.001499625 \\ 2.0001 & 2.00005 \end{array} \begin{array}{l|l} \mathrm{w} & \frac{\mathrm{w}-2}{\ln (\mathrm{w} / 2)} \\ \hline 1.9 & 1.949572575 \\ 1.99 & 1.994995823 \\ 1.9992 & 1.999599973 \\ 1.9999 & 1.99995 \end{array}



Practice 3:

\lim\limits_{x \rightarrow 0^{-}} f(x)=1
\lim\limits_{x \rightarrow 1^{-}} f(x)=1
\lim\limits_{x \rightarrow 2^{-}} f(x)=-1
\lim\limits_{x \rightarrow 3^{-}} f(x)=-1

\lim\limits_{x \rightarrow 0^{+}} f(x)=2
\lim\limits_{x \rightarrow 1^{+}} f(x)=1
\lim\limits_{x \rightarrow 2^{+}} f(x)=-1
\lim\limits_{x \rightarrow 3^{+}} f(x)=1

\lim\limits_{x \rightarrow 0} f(x) \, \text{does not exist}
\lim\limits_{x \rightarrow 1} f(x) 1
\lim\limits_{x \rightarrow 2} f(x)=-1
\lim\limits_{x \rightarrow 3} f(x) \, \text{does not exist}



Practice 4:

 \lim\limits_{x \rightarrow 1^{-}} f(x)=1
\lim\limits_{x \rightarrow 3^{-}} f(x)=3		 \lim\limits_{x \rightarrow 1^{+}} f(x)=1
\lim\limits_{x \rightarrow 3^{+}} f(x)=2 		 \lim\limits_{x \rightarrow 1} f(x)=1
\lim\limits_{x \rightarrow 3} f(x) \, \text{does not exist}