Practice Problems

Work through the odd-numbered problems 1-21. Once you have completed the problem set, check your answers.


1. Use the functions \mathrm{f} and \mathrm{g} defined by the graphs in Fig. 10 to determine the following limits.

(a) \lim\limits_{x \rightarrow 1}\{\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})\}

(b) \lim\limits_{x \rightarrow 1} f(x) \cdot g(x)

(c) \lim\limits_{x \rightarrow 1} f(x) / g(x)

(d) \lim\limits_{x \rightarrow 1} \mathrm{f}(\mathrm{g}(\mathrm{x}))

3. Use the function h defined by the graph in Fig. 11 to determine the following limits.

(a) \lim\limits_{x \rightarrow 2} \mathrm{~h}(2 \mathrm{x}-2)

(b) \lim\limits_{x \rightarrow 2}\{x+h(x)\}

(c) \lim\limits_{x \rightarrow 2} \mathrm{~h}(1+\mathrm{x})

(d) \lim\limits_{x \rightarrow 3} \mathrm{~h}(\mathrm{x} / 2)

5. Label the parts of the graph of f (Fig. 12) which are described by

(a) 2+\mathrm{h}

(b) \mathrm{f}(2)

(c) f(2+h)

(d) f(2+h)-f(2)

(e) \frac{f(2+h)-f(2)}{(2+h)-(2)}

(f) \frac{f(2-h)-f(2)}{(2-h)-(2)}

7. Use the function f defined by the graph in Fig. 14 to determine the following limits.

(a) \lim\limits_{x \rightarrow 1^{+}} f(x)

(b) \lim\limits_{x \rightarrow 1^{-}} f(x)

(c) \quad \lim\limits_{x \rightarrow 1} \mathrm{f}(\mathrm{x})

(d) \lim\limits_{x \rightarrow 3^{+}} \mathrm{f}(\mathrm{x})

(e) \lim\limits_{x \rightarrow 3^{-}} f(x)

(f) \lim\limits_{x \rightarrow 3} f(x)

(g) \lim\limits_{x \rightarrow-1^{+}} f(x)

(h) \lim\limits_{x \rightarrow-1^{-}} f(x)

(i) \lim\limits_{x \rightarrow-1} f(x)

9. The Lorentz Contraction Formula in relativity theory says the length \mathrm{L} of an object moving at \mathrm{v} miles per second with respect to an observer is \mathrm{L}=\mathrm{A} \cdot \sqrt{1-\frac{\mathrm{v}^{2}}{\mathrm{c}^{2}}} where \mathrm{c} is the speed of light (a constant).

a) Determine the "rest length" of the object (\mathrm{v}=0).

b) Determine \lim\limits_{v \rightarrow c^{-}} \mathrm{L}.

11. f(x)= \begin{cases}1 & \text { if } x < 1 \\ x & \text { if } 1 < x\end{cases} \quadand \quad g(x)=\left\{\begin{aligned} x & \text { if } x \neq 2 \\ 3 & \text { if } x=2 \end{aligned}\right.

(a) \lim\limits_{x \rightarrow 2}\{\mathrm{f}(\mathrm{x})+\mathrm{g}(\mathrm{x})\}

(b) \lim\limits_{x \rightarrow 2} f(x) / g(x)

(c) \lim\limits_{x \rightarrow 2} \mathrm{f}(\mathrm{g}(\mathrm{x}))

(d) \lim\limits_{x \rightarrow 0} \mathrm{~g}(\mathrm{x}) / \mathrm{f}(\mathrm{x})

(e) \quad \lim\limits_{x \rightarrow 1} \mathrm{f}(\mathrm{x}) / \mathrm{g}(\mathrm{x})

(f) \quad \lim\limits_{x \rightarrow 1} g(f(x))

Problems 13 and 15 require a calculator. 

13. (a) What does \lim\limits_{h \rightarrow 0} \frac{\cos (h)-1}{h} represent on the graph of \mathrm{y}=\cos (\mathrm{x}) ?

(It may help to recognize that \frac{\cos (\mathrm{h})-1}{\mathrm{~h}}=\frac{\cos (0+\mathrm{h})-\cos (0)}{\mathrm{h}})

(b) Graphically and using your calculator, determine \lim\limits_{h \rightarrow 0} \frac{\cos (h)-1}{h}.

15. Use your calculator (to generate a table of values) to help you estimate

(a) \lim\limits_{h \rightarrow 0} \frac{e^{h}-1}{h}

(b) \lim\limits_{c \rightarrow 0} \frac{\tan (1+c)-\tan (1)}{c}

(c) \lim\limits_{t \rightarrow 0} \frac{g(2+t)-g(2)}{t} when \mathrm{g}(\mathrm{t})=\mathrm{t}^{2}-5.

17. Describe the behavior of the function \mathrm{y}=\mathrm{f}(\mathrm{x}) in Fig. 16 at each integer using one of the phrases:

(a) "connected and smooth", (b) "connected with a corner",

(c) "not connected because of a simple hole which could be plugged by adding or moving one point", or

(d) "not connected because of a vertical jump which could not be plugged by moving one point".

19. This problem outlines the steps of a proof that \lim\limits_{\theta \rightarrow 0^{+}} \frac{\sin (\theta)}{\theta}=1

Statements (a)-(h) below refer to Fig. 18. Assume that 0 < \theta  < \frac{\pi}{2} \quad and justify why

each statement is true.

(a) Area of \Delta \mathrm{OPB}=\frac{1}{2} (base)(height) =\frac{1}{2} \sin (\theta).

(b) \frac{\text { area of the sector (the pie shaped region) } \mathrm{OPB}}{\text { area of the whole circle }}

=\frac{\text { angle defining sector } \mathrm{OPB}}{\text { angle of the whole circle }}=\frac{\theta}{2 \pi}

so ( area of the sector \mathrm{OPB})=\frac{\theta \pi}{2 \pi}=\frac{\theta}{2}.

(c) The line \mathrm{L} through the points (0,0) and \mathrm{P}=(\cos (\theta), \sin (\theta)) has slope \mathrm{m}=\frac{\sin (\theta)}{\cos (\theta)}, so \mathrm{C}=\left(1, \frac{\sin (\theta)}{\cos (\theta)}\right) and the area of \Delta \mathrm{OCB}=\frac{1}{2} (base)(height) =\frac{1}{2}(1) \frac{\sin (\theta)}{\cos (\theta)}

(d) Area of \Delta \mathrm{OPB} < area of sector \mathrm{OPB} < area of \triangle \mathrm{OCB}.

(e) \frac{1}{2} \sin (\theta) < \frac{\theta}{2} < \frac{1}{2}(1) \frac{\sin (\theta)}{\cos (\theta)} \quad and \sin (\theta) < \theta < \frac{\sin (\theta)}{\cos (\theta)}.

(f) 1 < \frac{\theta}{\sin (\theta)} < \frac{1}{\cos (\theta)} \quad and 1>\frac{\sin (\theta)}{\theta}>\cos (\theta)

(g) \lim\limits_{\theta \rightarrow 0^{+}} 1=1 and \lim\limits_{\theta \rightarrow 0^{+}} \cos (\theta)=1.

(h) \lim\limits_{\theta \rightarrow 0^{+}} \frac{\sin (\theta)}{\theta}=1.

21. Show that \quad \lim\limits_{x \rightarrow 0} \sin \left(\frac{1}{\mathrm{x}}\right) does not exist.

(Suggestion: Let \mathrm{f}(\mathrm{x})=\sin \left(\frac{1}{\mathrm{x}}\right) and pick \mathrm{a}_{\mathrm{n}}=\frac{1}{\mathrm{n} \pi} so \mathrm{f}\left(\mathrm{a}_{\mathrm{n}}\right)=\sin \left(\frac{1}{\mathrm{a}_{\mathrm{n}}}\right)=\sin
                            (\mathrm{n} \pi)=0 for every \mathrm{n}. Then pick b_{n}=\frac{1}{2 n \pi+\pi / 2} \quad so f\left(b_{n}\right)=\sin \left(\frac{1}{b_{n}}\right)=\sin \left(2 n \pi+\frac{\pi}{2}\right)=\sin \left(\frac{\pi}{2}\right)=1 for every \mathrm{n}).

Source: Dave Hoffman,
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