MHF 4UP
AP Unit Test/MCV4UP Unit 1 Test: Limits & Continuity (Version 1)
K
T
30
1.
C
8
A
8
Test Mark
Name: _____________________________________
8
y = f(x)
Use the graph of y = f(x) given at the right
to fill in the blanks below. [K-15]
(a) lim! 𝑓(𝑥)
_______
(b) lim" 𝑓(𝑥)
_______
(c) lim 𝑓(𝑥)
_______
(d) lim 𝑓(𝑥)
_______
(e) lim ! 𝑓(𝑥)
_______
(f) lim " 𝑓(𝑥)
_______
(g) lim 𝑓(𝑥) _______
(h) lim! 𝑓(𝑥)
_______
(i) lim" 𝑓(𝑥)
_______
(j) lim 𝑓(𝑥)
_______
(k)
_______
(l) lim" 𝑓(𝑥)
_______
(m) lim 𝑓(𝑥)
_______
!→#
!→#
!→#
!→$
!→%&
!→&
lim 𝑓(𝑥)
!→%'
!→%&
!→&
!→(
!→%&
!→&
!→(
(n) State all locations where f(x) is discontinuous on the interval (−∞, 8]: ____________________
[2 marks]
2.
Determine each of the following limits, if it exists. [K-6]
Only final answers are required; place them in the chart at the bottom of the question.
If the limit does not exist, state whether the limit approaches −∞ or ∞, if applicable.
Incorrect answer with rough work next to the question may earn you 0.5.
(a)
(d)
(a)
lim"
!→)
lim
!→'
! # % #+
(b)
!%)
#)! $ % +! # 0 -! % /
(e)
)! $ % &! 0 /
(b)
(c)
lim
(! % -)(! % /)
!→&!
lim
(c)
!%&
&
(f)
!→%' ! #
(d)
(e)
lim (𝑥 # − 7𝑥 + 3)
!→#
lim 2!
!→'
(f)
3.
Evaluate each of the following limits, if it exists. Show your work. [K-3, 3, 3]
(a) lim
!→&
! # % -! 0 /#
!#
%1
(b) lim
!→2
$
√! 0 #+ % +
!
4.
Evaluate the following limit, if it exists:
5.
Determine the value of 𝒌 such that 𝑓(𝑥) = &
%/
(c) lim % !! #% /
!→/
lim
!→+
|! % +| √! 0 )
!%+
5𝑥 ' − 4𝒌, 𝑥 ≤ 2
(
𝑥 − 𝒌𝑥, 𝑥 > 2
[T-5]
is continuous on (−∞, ∞). [T-3]
6.
If lim 𝑓(𝑥) = 3, determine the following limit using properties of limits. [C-4]
!→+
lim
!→+
7.
𝑓(𝑥) + 11
𝑥# − 4
Apply the Continuity Test to determine whether or not 𝑓 is continuous on the interval (−∞, ∞). [C-4]
𝑓(𝑥) = &
𝑥 ' + 5𝑥 − 8,
4) − 10,
𝑥>2
𝑥≤2
8(a)
Determine the average rate of change of 𝑓(𝑥) = −𝑥 # + 7𝑥 on the interval [−2, 3]. [A-2]
(b)
Determine the slope of the curve 𝑓(𝑥) = −𝑥 # + 7𝑥 at 𝑥 = 2. [A-4]
(c)
Determine the equation of the normal of 𝑓(𝑥) = −𝑥 # + 7𝑥 at 𝑥 = 2. [A-2]
AP TOPIC QUESTIONS
1.
Apply the Squeeze Theorem to show that lim
567 !
!→' #%
= 0. [4 marks]
10
2.
Evaluate the following limits, if they exist. Must show your work for (a). [6 marks – 3, 1, 1, 1]
(a)
lim
Incorrect answer with rough work next to the question may earn you 0.5.
!
(b)
lim 𝑥 & 𝑒 %!
___________
lim 𝑥 & 𝑒 %!
___________
!→%'
(c)
!→'
(d)
lim
#! * % &! # 0 /
!→%'
!%(
To end on a fun note…
A teacher explained the following limit to a student:
lim
$
!→#! !%#
= ∞
To see if the student understood the concept,
she asked the following similar question:
lim
$
!→&! !%&
Here was the student’s answer:
lim
$
!→&! !%&
=
5
!→2
567 +! 0 &!
___________