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Activity2.14Supplement

Subsection2.14.1Exercises

For Exercises 2.14.1.1–5, state the limit suggested by the values in the table and state whether or not the limit exists. The correct answer for Exercise 2.14.1.1 has been given to help you understand the instructions.

1
\(t\) \(\fe{g}{t}\)
\(-10{,}000\) \(99.97\)
\(-100{,}000\) \(999.997\)
\(-1{,}000{,}000\) \(9999.9997\)

Answer:\(\lim\limits_{t\to-\infty}\fe{g}{t}=\infty\text{.}\)

The limit does not exist.

2
\(x\) \(\fe{f}{x}\)
\(51{,}000\) \(-3.2\times10^{-5}\)
\(510{,}000\) \(-3.02\times10^{-7}\)
\(5{,}100{,}000\) \(-3.002\times10^{-9}\)
3
\(t\) \(\fe{z}{t}\)
\(0.33\) \(0.66\)
\(0.333\) \(0.666\)
\(0.3333\) \(0.6666\)
4
\(\theta\) \(\fe{g}{\theta}\)
\(-0.9\) \(2{,}999{,}990\)
\(-0.99\) \(2{,}999{,}999\)
\(-0.999\) \(2{,}999{,}999.9\)
5
\(t\) \(\fe{T}{t}\)
\(0.778\) \(29{,}990\)
\(0.7778\) \(299{,}999\)
\(0.77778\) \(2{,}999{,}999.9\)
6

Sketch onto Figure 2.14.1 a function, \(f\text{,}\) with the following properties. Your graph should include all of the features addressed in lab.

\begin{align*} \lim\limits_{x\to-4^{-}}\fe{f}{x}\amp=1\\ \lim\limits_{x\to-4^{+}}\fe{f}{x}\amp=-2\\ \lim\limits_{x\to3}\fe{f}{x}=\lim\limits_{x\to\infty}\fe{f}{x}\amp=-\infty \end{align*}

The only discontinuities on \(f\) are at \(-4\) and \(3\text{.}\) \(f\) has no \(x\)-intercepts. \(f\) is continuous from the right at \(-4\text{.}\) \(f\) has constant slope \(-2\) over \(\ointerval{-\infty}{-4}\text{.}\)

<<SVG image is unavailable, or your browser cannot render it>>

Figure2.14.1\(y=\fe{f}{x}\)
7

Sketch onto Figure 2.14.2 a function, \(f\text{,}\) with each of the properties stated below. Assume that there are no intercepts or discontinuities other than those directly implied by the given properties. Make sure that your graph includes all of the relevant features addressed in lab.

\begin{align*} \fe{f}{-2}\amp=0\\ \fe{f}{0}\amp=-1\\ \fe{f}{-4}\amp=5\\ \lim\limits_{x\to-\infty}\fe{f}{x}=\lim\limits_{x\to\infty}\fe{f}{x}\amp=3\\ \lim\limits_{x\to-4^{+}}\fe{f}{x}\amp=-2\\ \lim\limits_{x\to-4^{-}}\fe{f}{x}\amp=5\\ \lim\limits_{x\to3{-}}\fe{f}{x}\amp=-\infty\\ \lim\limits_{x\to3^{+}}\fe{f}{x}\amp=\infty \end{align*}

<<SVG image is unavailable, or your browser cannot render it>>

Figure2.14.2\(y=\fe{f}{x}\)
8

Determine all of the values of \(x\) where the function \(f\) (given below) has discontinuities. At each value where \(f\) has a discontinuity, determine if \(f\) is continuous from either the right or left at \(x\) and also state whether or not the discontinuity is removable.

\begin{equation*} \fe{f}{x}=\begin{cases}\frac{2\pi}{x}&x\leq3\\\fe{\sin}{\frac{2\pi}{x}}&3\lt x\lt 4\\\frac{\fe{\sin}{x}}{\fe{\sin}{x}}&4\lt x\leq7\\3-\frac{2x-8}{x-4}&x\gt7\end{cases} \end{equation*}
9

Determine the value(s) of \(k\) that make(s) the function \(g\) defined by \(\fe{g}{t}=\begin{cases}t^2+kt-k&t\geq3\\t^2-4k&t\lt 3\end{cases}\) continuous over \(\ointerval{-\infty}{\infty}\text{.}\)

Determine the appropriate symbol to write after an equal sign following each of the given limits. In each case, the appropriate symbol is either a real number, \(\infty\text{,}\) or \(-\infty\text{.}\) Also, state whether or not each limit exists and if the limit exists prove its existence (and value) by applying the appropriate limit laws. The Rational Limit Forms table in Appendix A summarizes strategies to be employed based upon the initial form of the limit.

10

\(\lim\limits_{x\to4^{-}}\left(5-\dfrac{1}{x-4}\right)\)

11

\(\lim\limits_{x\to\infty}\dfrac{e^{\sfrac{2}{x}}}{e^{\sfrac{1}{x}}}\)

12

\(\lim\limits_{x\to2^{+}}\dfrac{x^2-4}{x^2+4}\)

13

\(\lim\limits_{x\to2^{+}}\dfrac{x^2-4}{x^2-4x+4}\)

14

\(\lim\limits_{x\to\infty}\dfrac{\fe{\ln}{x}+\fe{\ln}{x^6}}{7\fe{\ln}{x^2}}\)

15

\(\lim\limits_{x\to-\infty}\dfrac{3x^3+2x}{3x-2x^3}\)

16

\(\lim\limits_{x\to\infty}\fe{\sin}{\dfrac{\pi e^{3x}}{2e^x+4e^{3x}}}\)

17

\(\lim\limits_{x\to\infty}\dfrac{\fe{\ln}{\sfrac{1}{x}}}{\fe{\ln}{\sfrac{x}{x}}}\)

18

\(\lim\limits_{x\to5}\sqrt{\dfrac{x^2-12x+35}{5-x}}\)

19

\(\lim\limits_{h\to0}\dfrac{4(3+h)^2-5(3+h)-21}{h}\)

20

\(\lim\limits_{h\to0}\dfrac{5h^2+3}{2-3h^2}\)

21

\(\lim\limits_{h\to0}\dfrac{\sqrt{9-h}-3}{h}\)

22

\(\lim\limits_{\theta\to\frac{\pi}{2}}\dfrac{\fe{\sin}{\theta+\frac{\pi}{2}}}{\fe{\sin}{2\theta+\pi}}\)

23

\(\lim\limits_{x\to0^{+}}\dfrac{\fe{\ln}{x^e}}{\fe{\ln}{e^x}}\)

Draw sketches of the curves \(y=e^x\text{,}\) \(y=e^{-x}\text{,}\) \(y=\fe{\ln}{x}\text{,}\) and \(y=\frac{1}{x}\text{.}\) Note the coordinates of any and all intercepts. This is something you should be able to do from memory/intuition. If you cannot already do so, spend some time reviewing whatever you need to review so that you can do so. Once you have the graphs drawn, fill in each of the blanks below and decide whether or not each limit exists. The appropriate symbol for some of the blanks is either \(\infty\) or \(-\infty\text{.}\)

24

\(\lim\limits_{x\to\infty}e^x=\underline{\qquad}\,(\text{exists?})\)

25

\(\lim\limits_{x\to-\infty}e^x=\underline{\qquad}\,(\text{exists?})\)

26

\(\lim\limits_{x\to0}e^x=\underline{\qquad}\,(\text{exists?})\)

27

\(\lim\limits_{x\to\infty}e^{-x}=\underline{\qquad}\,(\text{exists?})\)

28

\(\lim\limits_{x\to-\infty}e^{-x}=\underline{\qquad}\,(\text{exists?})\)

29

\(\lim\limits_{x\to0}e^{-x}=\underline{\qquad}\,(\text{exists?})\)

30

\(\lim\limits_{x\to\infty}\fe{\ln}{x}=\underline{\qquad}\,(\text{exists?})\)

31

\(\lim\limits_{x\to1}\fe{\ln}{x}=\underline{\qquad}\,(\text{exists?})\)

32

\(\lim\limits_{x\to0^{+}}\fe{\ln}{x}=\underline{\qquad}\,(\text{exists?})\)

33

\(\lim\limits_{x\to\infty}\dfrac{1}{x}=\underline{\qquad}\,(\text{exists?})\)

34

\(\lim\limits_{x\to-\infty}\dfrac{1}{x}=\underline{\qquad}\,(\text{exists?})\)

35

\(\lim\limits_{x\to0^{+}}\dfrac{1}{x}=\underline{\qquad}\,(\text{exists?})\)

36

\(\lim\limits_{x\to0^{-}}\dfrac{1}{x}=\underline{\qquad}\,(\text{exists?})\)

37

\(\lim\limits_{x\to\infty}e^{\sfrac{1}{x}}=\underline{\qquad}\,(\text{exists?})\)

38

\(\lim\limits_{x\to\infty}\dfrac{1}{e^x}=\underline{\qquad}\,(\text{exists?})\)

39

\(\lim\limits_{x\to-\infty}\dfrac{1}{e^x}=\underline{\qquad}\,(\text{exists?})\)

40

\(\lim\limits_{x\to\infty}\dfrac{1}{e^{-x}}=\underline{\qquad}\,(\text{exists?})\)

41

\(\lim\limits_{x\to-\infty}\dfrac{1}{e^{-x}}=\underline{\qquad}\,(\text{exists?})\)