 Precalculus

# Review Exercises

PrecalculusReview Exercises

### Review Exercises

##### Finding Limits: A Numerical and Graphical Approach

For the following exercises, use Figure 1.

Figure 1
1.

$lim x→ −1 + f(x) lim x→ −1 + f(x)$

2.

$lim x→ −1 − f(x) lim x→ −1 − f(x)$

3.

$lim x→−1 f(x) lim x→−1 f(x)$

4.

$lim x→3 f(x) lim x→3 f(x)$

5.

At what values of $x x$ is the function discontinuous? What condition of continuity is violated?

6.

Using Table 1, estimate $lim x→0 f(x). lim x→0 f(x).$

 $xx$ $F(x) F(x)$ −0.1 2.875 −0.01 2.92 −0.001 2.998 0 Undefined 0.001 2.9987 0.01 2.865 0.1 2.78145 0.15 2.678
Table 1

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $x x$ approaches $a. a.$ If the function has limit as $x x$ approaches $a, a,$ state it. If not, discuss why there is no limit.

7.

8.

9.

##### Finding Limits: Properties of Limits

For the following exercises, find the limits if $lim x→c f( x )=−3 lim x→c f( x )=−3$ and $lim x→c g( x )=5. lim x→c g( x )=5.$

10.

$lim x→c ( f(x)+g(x) ) lim x→c ( f(x)+g(x) )$

11.

$lim x→c f(x) g(x) lim x→c f(x) g(x)$

12.

$lim x→c ( f(x)⋅g(x) ) lim x→c ( f(x)⋅g(x) )$

13.

14.

15.

$lim x→ 3 + ( 3x−[x] ) lim x→ 3 + ( 3x−[x] )$

For the following exercises, evaluate the limits using algebraic techniques.

16.

$lim h→0 ( ( h+6 ) 2 −36 h ) lim h→0 ( ( h+6 ) 2 −36 h )$

17.

$lim x→25 ( x 2 −625 x −5 ) lim x→25 ( x 2 −625 x −5 )$

18.

$lim x→1 ( − x 2 −9x x ) lim x→1 ( − x 2 −9x x )$

19.

$lim x→4 7− 12x+1 x−4 lim x→4 7− 12x+1 x−4$

20.

$lim x→−3 ( 1 3 + 1 x 3+x ) lim x→−3 ( 1 3 + 1 x 3+x )$

##### Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at $x=a. x=a.$ If not, describe the behavior of the graph of the function at $x=a. x=a.$

21.

$f(x)= −2 x−4 ;a=4 f(x)= −2 x−4 ;a=4$

22.

$f(x)= −2 ( x−4 ) 2 ;a=4 f(x)= −2 ( x−4 ) 2 ;a=4$

23.

$f(x)= −x x 2 −x−6 ;a=3 f(x)= −x x 2 −x−6 ;a=3$

24.

$f(x)= 6 x 2 +23x+20 4 x 2 −25 ;a=− 5 2 f(x)= 6 x 2 +23x+20 4 x 2 −25 ;a=− 5 2$

25.

$f(x)= x −3 9−x ;a=9 f(x)= x −3 9−x ;a=9$

For the following exercises, determine where the given function $f(x) f(x)$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

26.

$f(x)= x 2 −2x−15 f(x)= x 2 −2x−15$

27.

$f(x)= x 2 −2x−15 x−5 f(x)= x 2 −2x−15 x−5$

28.

$f(x)= x 2 −2x x 2 −4x+4 f(x)= x 2 −2x x 2 −4x+4$

29.

$f(x)= x 3 −125 2 x 2 −12x+10 f(x)= x 3 −125 2 x 2 −12x+10$

30.

$f(x)= x 2 − 1 x 2−x f(x)= x 2 − 1 x 2−x$

31.

$f(x)= x+2 x 2 −3x−10 f(x)= x+2 x 2 −3x−10$

32.

$f(x)= x+2 x 3 +8 f(x)= x+2 x 3 +8$

##### Derivatives

For the following exercises, find the average rate of change $f(x+h)−f(x) h . f(x+h)−f(x) h .$

33.

$f(x)=3x+2 f(x)=3x+2$

34.

$f(x)=5 f(x)=5$

35.

$f(x)= 1 x+1 f(x)= 1 x+1$

36.

$f(x)=ln(x) f(x)=ln(x)$

37.

$f(x)= e 2x f(x)= e 2x$

For the following exercises, find the derivative of the function.

38.

$f(x)=4x−6 f(x)=4x−6$

39.

$f(x)=5 x 2 −3x f(x)=5 x 2 −3x$

40.

Find the equation of the tangent line to the graph of $f( x ) f( x )$ at the indicated $x x$ value.

$f(x)=− x 3 +4x f(x)=− x 3 +4x$; $x=2. x=2.$

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

41.

$f(x)= x | x | f(x)= x | x |$

42.

Given that the volume of a right circular cone is $V= 1 3 π r 2 h V= 1 3 π r 2 h$ and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of $π π$

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