Math 165: Calculus 1
Fall 2025 Quiz 2
Name
TA Name
Recitation Time
Instructor Name (Boynton or Littmann)
Directions. Work all nine of the problems without books, notes, or calculators. Show all work and justify
all answers for full credit. It is possible to earn 110/100 on this quiz. Good luck and just do the best you
can.
Exercise
Score
1
=20
2
=10
3
=10
4
=10
5
=10
6
=10
7
=20
8
=10
9
=10
TOTAL
=100
1
1. (20) Consider the function y = f (x) displayed below.
(a) Write down all values of x for which the function f has a removable discontinuity.
x=
(b) Write down all values of x for which the function f has a jump discontinuity.
x=
(c) Write down all values of x for which the function f has an in…nite discontinuity.
x=
(d) In order to make f continuous at x =
3, we should rede…ne the function so that
f ( 3) =
(e) At the point where x = 0, the function f is continuous from the
answer choice.
RIGHT
LEFT
2
. Circle the best
2. (10) Evaluate the following limit if it exists. Show all work in the space provided. Enter your
simpli…ed answer in the square bracketed space. If the limit is in…nite, answer with either +1 or
1: If the limit does not exist, write "DNE" in the answer box.
lim tan 1
x!2
x2
5x + 6
x 2
3
=
3. (10) Evaluate the following limit if it exists. Show all work in the space provided. Enter your
simpli…ed answer in the square bracketed space. If the limit is in…nite, answer with either +1 or
1: If the limit does not exist, write "DNE" in the answer box.
3x tan(x)
=
x!0 sin2 (2x)
lim
4
4. (10) Evaluate the limit if it exists. Show all work in the space provided. Enter your simpli…ed
answer in the square bracketed space. If the limit exists and is in…nite, answer with either +1 or
1: If the limit does not exist, write DNE.
p
3
x6 1
lim p
=
x! 1
4x4 + 3x
5
5. (10) Evaluate the limit if it exists. Show all work in the space provided. Enter your simpli…ed
answer in the square bracketed space. If the limit exists and is in…nite, answer with either +1 or
1: If the limit does not exist, write DNE.
lim x2
x!0
Hint: Recall that ~
1
b~c
3
+ cos(x)
x2
165 =
~ no matter what ~ is.
6
6. (10) Consider the function
ln (x + 3) + 2
:
x 1
Find all points at which the function f is continuous. Enter your answer in interval notation.
f (x) =
AN S =
7
7. (20) In each case, circle the best description of the given function f at the point where x = a: Justify
your answer with a relevant limit computation.
(a)
f (x) =
(i)
(ii)
(iii)
(iv)
x2 1
x 1
x
1
x<1
x 1
at x = 1
f (x) is continuous.
f (x) has a removable discontinuity.
f (x) has a jump discontinuity
f (x) has an in…nite discontinuity
(b)
f (x) =
(i)
(ii)
(iii)
(iv)
(
(
p x
x+4 2
1
x 6= 0
x=0
at x = 0
f (x) is continuous.
f (x) has a removable discontinuity.
f (x) has a jump discontinuity
f (x) has an in…nite discontinuity
8
(c)
8 2
1 1
< x tan ( x )
0
f (x) =
: 1 cos(x)
x<0
x=0
x>0
x
(i)
(ii)
(iii)
(iv)
at x = 0
f (x) is continuous.
f (x) has a removable discontinuity.
f (x) has a jump discontinuity
f (x) has an in…nite discontinuity
(d)
f (x) =
(i)
(ii)
(iii)
(iv)
1
jx
1
2j
x
2
at x = 2
f (x) is continuous.
f (x) has a removable discontinuity.
f (x) has a jump discontinuity
f (x) has an in…nite discontinuity
9
8. (10) Use the Intermediate Value Theorem to explain why the equation 3
number solution.
10
p
x=
p
x + 2 has a real
9. (10) True or False? Circle the best answer choice.
(a) If f is continuous at x = 2, then lim f (x) = f (2):
x!2
TRUE
FALSE
(b) An elementary function is continuous everywhere on its domain.
TRUE
FALSE
(c) A function f can have a removable discontinuity at x = a and be right continuous at x = a:
TRUE
FALSE
(d) A function f can have an in…nite discontinuity at x = a and be right continuous at x = a:
TRUE
(e) If l(x)
f (x)
FALSE
u(x) for all x near 1, lim l(x) =
x!1
1, and lim u(x) = 1, then lim f (x) does not
x!1
x!1
exist.
TRUE
FALSE
(f) A function f can have at most one horizontal asymptote.
TRUE
FALSE
(g) If f has a removable discontinuity at x = a and limx!a f (x) = L, then
g(x) =
f (x)
L
x 6= a
x=a
also has a removable discontinuity at x = a:
TRUE
FALSE
(h) If f is continuous at x = 5 and f (5) = 3, then lim f (4x2
x!2
TRUE
11) = 3:
FALSE
(i) If jf j is continuous at x = a, then f is also continuous at x = a:
TRUE
FALSE
(j) If the function f is not continuous on the interval [ 1; 1], then the conclusion of the Intermediate
Value Theorem does not hold. In other words, we are guaranteed to …nd a real number M
properly between f ( 1) and f (1) such that f (c) 6= M no matter what c we pick in ( 1; 1):
TRUE
FALSE
11