Practice Problems

Work through the odd-numbered problems 1-19. Once you have completed the problem set, check your answers.


1. Use the graph in Fig. 10 to determine the following limits.

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

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

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

(d) \lim\limits_{x \rightarrow 4} f(x)

3. Use the graph in Fig. 12 to determine the following limits.

(a) \lim\limits_{x \rightarrow 1} \mathrm{f}(2 \mathrm{x})

(b) \lim\limits_{x \rightarrow 2} \mathrm{f}(\mathrm{x}-1)

(c) \lim\limits_{x \rightarrow 3} \mathrm{f}(2 \mathrm{x}-5)

(d) \lim\limits_{x \rightarrow 0} \mathrm{f}(4+\mathrm{x})

5. Evaluate  (a) \lim\limits_{x \rightarrow 1} \frac{x^{2}+3 x+3}{x-2}      (b) \lim\limits_{x \rightarrow 2} \frac{x^{2}+3 x+3}{x-2}

7. Evaluate  (a) \lim\limits_{x \rightarrow 1} \frac{\cos (x)}{x}     (b) \lim\limits_{x \rightarrow \pi} \frac{\cos (x)}{x}      (c) \lim\limits_{x \rightarrow-1} \frac{\cos (x)}{x}

9. Evaluate  (a) \lim\limits_{x \rightarrow 0^{-}}|x|     (b) \lim\limits_{x \rightarrow 0^{+}}|x|      (c) \lim\limits_{x \rightarrow 0}|x|

11. Evaluate  (a) \lim\limits_{x \rightarrow 5}|x-5|     (b) \lim\limits_{x \rightarrow 3} \frac{|x-5|}{x-5}      (c) \lim\limits_{x \rightarrow 5} \frac{|x-5|}{x-5}

13.  g(x)= \begin{cases}1 & \text { if } x \leq 2 \\ 8 / x & \text { if } 2 < x < 4 \qquad \\ 6-x & \text { if } 4 < x \end{cases}         . Find the one and two-sided limits of \mathrm{g} as \mathrm{x} \rightarrow 1,2,4, and \mathrm{5}.

In problems 15 and 17, use a calculator or computer to get approximate answers accurate to 2 decimal places.

15. (a) \lim\limits_{x \rightarrow 0} \frac{3^{x}-1}{x}     (b) \lim\limits_{x \rightarrow 1} \frac{\ln (x)}{x-1}

17. (a) \lim\limits_{x \rightarrow 16} \frac{\sqrt{x}-4}{x-16}     (b) \lim\limits_{x \rightarrow 0} \frac{\sin (7 x)}{2 x}

19. Define \mathrm{A}(\mathrm{x}) to be the area bounded by the \mathrm{x} and \mathrm{y} axes, the line \mathrm{y}=\frac{1}{2} \mathrm{x}+2, and the vertical line at \mathrm{x}. (Fig. 15). For example, \mathrm{A}(4)=12.

a) Evaluate \mathrm{A}(0), \mathrm{A}(1), \mathrm{A}(2), and \mathrm{A}(3).

b) Graph y=A(x) for 0 \leq x \leq 4.

c) What area does A(3)-A(1) represent?

Source: Dale Hoffman,
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