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Limits and Continuity Exam Review 1
The Calculus Page 1 of 4
Find the following limits without using a graphing calculator or making tables.
#1) 52
15lim2
5 ��
o xx
x
#2) h
xhhxh
22
0
410lim �o
Answer each question concerning piecewise functions.
#3) 𝑓(𝑥) = {− 15
𝑥 + 3, 𝑖𝑓 𝑥 < 52𝑥 − 8, 𝑖𝑓 𝑥 ≥ 5
a.
�o)(lim
5xf
x
b. �o
)(lim5
xfx
c. o
)(lim5
xfx
#4) 𝑓(𝑥) = { 4𝑥 + 5, 𝑖𝑓 𝑥 < 1−2𝑥 + 7, 𝑖𝑓 𝑥 ≥ 1
a.
�o)(lim
1xf
x
b. �o
)(lim1
xfx
c. o
)(lim1
xfx
Limits and Continuity Exam Review 1
The Calculus Page 2 of 4
#5) For the following piecewise function:
𝑓(𝑥) = {13 𝑥 + 2, 𝑖𝑓 𝑥 < −3−2𝑥 − 5, 𝑖𝑓 𝑥 ≥ −3
a. Draw its graph
b. Find the limits as x approaches -3 from the left.
c. Find the limits as x approaches -3 from the right.
d. Is it continuous at x = -3? If not, why?
#6) Find each limit. Assume that each limit that does exist is an integer. (There is no work to be shown)
a. �o
)(lim1
xfx
b. ��o
)(lim2
xfx
c. �o
)(lim4
xfx
d. o
)(lim6
xfx
Limits and Continuity Exam Review 1
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#7) Find the equation for the tangent line to the curve 𝑓(𝑥) = 3𝑥2 − 3𝑥 + 1 at x = 0. Write your equation in slope-intercept form. (You must use the definition of derivative to find the slope.)
Given the graph of a function, sketch in the graph of its derivative function. #8)
Limits and Continuity Exam Review 1
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#9) Often times, problems will ask for the derivative without using the word “derivative”. We have learned two interpretations of a derivative. What are these two interpretations? #10) 13)3(lim 2
4 �
ox
x is read the “limit of 𝑥2 − 3,
as x approaches 4, is equal to 13.” Use sentences and graphs to illustrate the meaning of said statement.
#11) Give 2 specific scenarios of when a limit would not exist and explain why. You may use graphs to illustrate your point. Scenario #1: Scenario #2:
Derivative means slope of a tangent line. It also means instantaneous rate of change.