Practice Problems

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, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2011/11/2-2FunctionLimit.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. (a) $2$   (b) $1$   (c) $\text{DNE}$ (does not exist) (d) $1$

3. (a) $1$   (b) $-1$   (c) $-1$   (d) $2$

5. (a) $-7$   (b) (13/0) $\text{DNE}$

7. (a) $0.54$ (remember, we are using radian mode) (b) $-0.318$ (c) $-0.54$

9. (a) $0$   (b) $0$   (c) $0$

11. (a) $0$   (b) $-1$ (c) $\text{DNE}$

13. 

15. (a) $1.0986$   (b) $1$

17. (a) $0.125$  (b) $3.5$

19. (a) $\mathrm{A}(0)=0, \mathrm{~A}(1)=2.25, \mathrm{~A}(2)=5, \quad \mathrm{~A}(3)=8.25$

(b) $\mathrm{A}(\mathrm{x})=2 \mathrm{x}+\mathrm{x}^{2} / 4$

(c) the area of the region bounded below by the $x$-axis, above by the line $y=x / 2+2$, on the left by the vertical line $x=1$, and on the right by the vertical line $x=3$.