Limits Review for Quiz
Directions: Find each limit.
1.
5.
x
lim
2. lim
x2 3x 2
3 x  3
x 0
x 3  4x
 5x  2
lim 9  x (Why didn’t I ask for the right hand limit as well?)
6. lim
x1
x9
x2  1
x0 x  1
7. a. lim
8. lim
(x  1) 5
x1
x2  1
3x5  4x
x 5  6x5
b. lim
1
1

4. lim 2  x 2
x
x 0
tan 2 x
3. lim
x
x0
3x7  4x
x 5  9x5
x2  x
x2  1
3x5  4x
x 5  6x9
c. lim
d. lim
(Find the left and right hand limits as well as the overall limit)
x2
x1 1  x
9. lim
10.
x
x 1 x  1
lim
1  cos 2 x
x
x0
11. lim
12.
sin x
x  2 x
lim
(x  1) 2  1
x 0
x
13. lim
14.
x3  8
x2 x  2
lim
15.
lim
x1
x 4  2x 3  15x 2  32x  16
x 2  2x  1
2

bx  1 , x  -2
16. Let g(x)  
. What value of b makes g(x) continuous everywhere?
x  -2

x,
 x2  9
,x 3

17. Let g(x)   x  3
. What value of a makes g(x) continuous everywhere?
a,
x 3

 1
 x  3, x  2
18. If g(x)   2
, determine lim g(x) . Justify your answer.
x2
3x  1,
x2
19. Describe any discontinuities in the graph of h(x) 
2
x 3
x  5x  24
. (Be sure to state if they are removable or
nonremovable)

 2x  7 , x  2
20. Find the value of k that makes f(x) continuous, given f(x)  
2

x  x  k, x  2
21. Use the given graph to answer the questions.
a. lim g(x) 
b. lim g(x) 
x3
x1
5
g(x)
4
c. g(1)
d. g(-1)
e. g(6)
3
2
f.
1
-2
-1
1
2
3
4
5
6
7
lim g(x) 
g.
lim g(x) 
i. lim g(x) 
x6
lim g(x) 
x1
8
-1
-2
h.
x1
x0
-3
-4
Be sure you can justify your limits using left-hand and right hand limits.
j. Use the graph above and properties of limits to determine:
lim [g(x)  2] 
x3
and lim [2g(x)] 
x1
h. At which x-values in the graph of g(x) above discontinuous? (Classify each as removable or nonremovable and be
sure you can tell which of the three conditions of continuity fail)
Some other things that you should study/memorize. (You don’t have to write these down)
22. What it means for a limit to exist. What a limit is. (Be sure not to use “it” in your
explanations)
23. The two common trig limits
24. 3 conditions for continuity
25. What indeterminate form is and what it tells us about the graph or the function?
26. How can you tell if a limit is going to be unbounded? Once you know, how do you know in
which direction it is going?
27.Common values for sin(x) and cos(x).
28. Solve each of the following limits – you can make blueberry pancakes when you are done
 3x  23
x4 2x  3
_____egg(s)
x 2  7 x  12
x4
x4
_____cup(s) all-purpose flour*
lim
lim
 x 2  11x  28
lim
x 4
x 2  4x
x
lim sin  
x
6
x
lim 2 tan 

2
x
2
_____cup(s) milk
______cup(s) fresh or frozen
blueberries (thawed and well drained)
_____tablespoon(s) shortening (or
vegetable oil)
 x 2  7 x  12
x 3
x 3
_____tablespoon(s) sugar
2x 2  7 x  5
x  1
x 1
_____teaspoon(s) baking powder
lim
lim
lim
x 0
1  x 1
x
_____teaspoon(s) salt
Beat egg with hand beater until fluffy;
beat in remaining ingredients just until
smooth. Grease heated griddle if
necessary. (To test griddle, sprinkle
with a few drops of water. If bubbles
skitter around, heat is just right.)
Pour about 3 tablespoons batter from
tip of large spoon or from pitcher onto
hot griddle. Cook pancakes until
puffed and dry around the edges.
Turn and cook other sides until golden
brown.
Recipe makes about nine 4-inch
pancakes.
“The next part of this recipe will involve some calculus”
REVIEW 1.1 ANSWERS!!!
*If using self-rising flour, omit baking powder
and salt.
1. 2 3
2.
8
7
3. 0
4. 
1
4
5. 0, I did not ask for the right hand limit because the domain of this function is (,9] , so there is not right
hand limit because there is no graph to the right of 9.
1
6.
7. a. -1
b. -1/2
c. -∞
2
d. 0
8. Right =  , Left = -  , Overall DNE
9. Right = -  , Left =  , Overall DNE
10. - 
11. 0
12.
2

13. 2
14. 12
15. -15
3
17. 6
4
18. DNE b/c lim g(x)  2  lim g(x)  5
16. 
x2
x2
19. Removable at x = 3 and nonremovable at x = -8
 2  11
20.
21. a. 0
g. 4
j.
b. 2
h. DNE
c. 3
d. 4
e. 3
f. 3
i. 0
2 and 4
k. nonremovable at x = -1, removable at x = 1, nonremovable at x = 6
28. 1, 1, ¾, ½, 2, 1, 3, ½
JUST REMEMBER I AM NOT PERFECT SO THERE COULD BE TYPOS/MISTAKES
STUDY YOUR NOTES, HW, WARMUPS AND QUIZZES – EVERYTHING IS
FAIR GAME!!
Have your binders ready when you walk into class.