MA005: Calculus I

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

(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, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-2.3-Properties-of-Limits.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.