Section 1.6 Homework

 

1. Given that

find the limits, if they exist. (If an answer does not exist, enter DNE.)

 

 

$a. \underset{x\to 3}{\lim }\left[f\right(x) + 2g(x\left)\right]$

 

$b. \underset{x\to 3}{\lim }{\left[g\right(x\left)\right]}^3$

 

$c. \underset{x\to 3}{\lim }\sqrt{f\left(x\right)}$

 

$\underset{ x\to 3}{d. \lim }\frac{5f\left(x\right)}{g\left(x\right)}$

 

$e. \underset{x\to 3}{\lim }\frac{g\left(x\right)}{h\left(x\right)}$

 

f. 

$\underset{x\to 3}{\lim }\frac{g\left(x\right)h\left(x\right)}{f\left(x\right)}$

 

 

2. The graphs of f and g are given. Use them to evaluate each limit, if it exists. (If an answer does not exist, enter DNE.)

$a. \underset{x\to 2}{\lim }\left[f\right(x)+g(x\left)\right]$

 

 

$\underset{ x\to 0}{b. \lim }\left[f\right(x) – g(x\left)\right]$

$c. \underset{x\to –1}{\lim }\left[f\right(x\left)g\right(x\left)\right]$

 

 

$d. \underset{x\to 3}{\mathrm{li}m}\frac{f\left(x\right)}{g\left(x\right)}$

 

 

$e. \underset{x\to 2}{\lim }\left[x^{2}f\right(x\left)\right]$

 

$f. f(–1) + \underset{x\to –1}{\lim }g\left(x\right)$

 

 

3. Evaluate the limit using the appropriate Limit Law(s). (If an answer does not exist, enter DNE.)

 

 

4.  Evaluate the limit using the appropriate Limit Law(s). (If an answer does not exist, enter DNE.)

 

5.   Evaluate the limit using the appropriate Limit Law(s). (If an answer does not exist, enter DNE.)

 

6.  Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)

 

 

7. Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)

 

8.  Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)

 

9. Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)

 

 

10. Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)

 

 

11.  Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)

 

 

12. Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)

 

13.  Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.)

 

14.  The signum (or sign) function, denoted by sgn, is defined by

(a) Sketch the graph of this function.

 

 

(b) Find each of the following limits. (If an answer does not exist, enter DNE.)

$\left(i\right) \underset{x\to 0^{+}}{\lim }sgn x$

 

$\left(ii\right) \underset{x\to 0^{–}}{\lim }sgn x$

 

$\left(iii\right) \underset{x\to 0}{\lim }sgn x$

 

$\left(iv\right) \underset{x\to 0}{\lim } \left|sgn x\right|$

 

 

 

15. 

$Let g\left(x\right) = \frac{x^2+x–12}{\left|x–3\right|}$

(a) Find the following limits. (If an answer does not exist, enter DNE.)

 

$\left(i\right) \underset{x\to 3^{+}}{\lim }g\left(x\right)$

 

$(ii.) \underset{x\to 3^{–}}{\lim }g\left(x\right)$

 

(b) 

$Does \underset{x\to 3}{\lim }g\left(x\right) exist?$

 

(c) Sketch the graph of g.

 

 

16. Let

 

(a) Find the following limits. (If an answer does not exist, enter DNE.)

$\underset{x\to 1^{–}}{\lim }f\left(x\right) =$

 

$\underset{x\to 1^{+}}{\lim }f\left(x\right) =$

 

(b) Does

$\underset{x\to 1}{\lim }f\left(x\right) exist?$

 

(c) Sketch the graph of f.