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Math Problems
Calculus
Find derivatives of logarithmic functions
Find
lim
x
→
π
2
cot
2
(
x
)
1
−
sin
(
x
)
\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cot ^{2}(x)}{1-\sin (x)}
lim
x
→
2
π
1
−
s
i
n
(
x
)
c
o
t
2
(
x
)
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
−
π
2
-\frac{\pi}{2}
−
2
π
\newline
(C)
2
2
2
\newline
(D) The limit doesn't exist
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Find
lim
x
→
π
2
sin
(
2
x
)
cos
(
x
)
\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin (2 x)}{\cos (x)}
lim
x
→
2
π
c
o
s
(
x
)
s
i
n
(
2
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
\frac{1}{2}
2
1
\newline
(B)
1
1
1
\newline
(C)
2
2
2
\newline
(D) The limit doesn't exist
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Find
lim
x
→
0
sin
(
x
)
sin
(
2
x
)
\lim _{x \rightarrow 0} \frac{\sin (x)}{\sin (2 x)}
lim
x
→
0
s
i
n
(
2
x
)
s
i
n
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
\frac{1}{2}
2
1
\newline
(B)
1
1
1
\newline
(C)
2
2
2
\newline
(D) The limit doesn't exist
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Find
lim
x
→
3
x
−
3
4
x
+
4
−
4
\lim _{x \rightarrow 3} \frac{x-3}{\sqrt{4 x+4}-4}
lim
x
→
3
4
x
+
4
−
4
x
−
3
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
4
-4
−
4
\newline
(B)
1
1
1
\newline
(C)
2
2
2
\newline
(D) The limit doesn't exist
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Find
lim
x
→
−
4
7
x
+
28
x
2
+
x
−
12
\lim _{x \rightarrow-4} \frac{7 x+28}{x^{2}+x-12}
lim
x
→
−
4
x
2
+
x
−
12
7
x
+
28
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
1
1
\newline
(B)
7
7
7
\newline
(C)
−
1
-1
−
1
\newline
(D) The limit doesn't exist
Get tutor help
Find
lim
x
→
−
3
x
+
3
4
−
2
x
+
22
\lim _{x \rightarrow-3} \frac{x+3}{4-\sqrt{2 x+22}}
lim
x
→
−
3
4
−
2
x
+
22
x
+
3
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
3
-3
−
3
\newline
(B)
−
4
-4
−
4
\newline
(C)
−
3
4
-\frac{3}{4}
−
4
3
\newline
(D) The limit doesn't exist
Get tutor help
Find
lim
x
→
1
5
x
+
4
−
3
x
−
1
\lim _{x \rightarrow 1} \frac{\sqrt{5 x+4}-3}{x-1}
lim
x
→
1
x
−
1
5
x
+
4
−
3
.
\newline
Choose
1
1
1
answer:
\newline
(A)
3
5
\frac{3}{5}
5
3
\newline
(B)
5
6
\frac{5}{6}
6
5
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist
Get tutor help
Find
lim
x
→
−
2
x
3
+
3
x
2
+
2
x
x
+
2
\lim _{x \rightarrow-2} \frac{x^{3}+3 x^{2}+2 x}{x+2}
lim
x
→
−
2
x
+
2
x
3
+
3
x
2
+
2
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
6
6
6
\newline
(B)
0
0
0
\newline
(C)
2
2
2
\newline
(D) The limit doesn't exist
Get tutor help
Find
lim
θ
→
π
2
tan
2
(
θ
)
[
1
−
sin
(
θ
)
]
\lim_{\theta \rightarrow \frac{\pi}{2}} \tan ^{2}(\theta)[1-\sin (\theta)]
lim
θ
→
2
π
tan
2
(
θ
)
[
1
−
sin
(
θ
)]
.
\newline
Choose
1
1
1
answer:
\newline
(A)
0
0
0
\newline
(B)
1
2
\frac{1}{2}
2
1
\newline
(C)
−
2
-2
−
2
\newline
(D) The limit doesn't exist
Get tutor help
Find
lim
θ
→
π
2
sin
2
(
2
θ
)
1
−
sin
2
(
θ
)
\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{\sin ^{2}(2 \theta)}{1-\sin ^{2}(\theta)}
lim
θ
→
2
π
1
−
s
i
n
2
(
θ
)
s
i
n
2
(
2
θ
)
\newline
Choose
1
1
1
answer:
\newline
(A)
1
1
1
\newline
(B)
2
2
2
\newline
(C)
4
4
4
\newline
(D) The limit doesn't exist
Get tutor help
lim
x
→
0
cot
(
x
)
=
?
\lim _{x \rightarrow 0} \cot (x)=?
x
→
0
lim
cot
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
Get tutor help
f
(
x
)
=
{
x
2
for
x
≤
0
ln
(
x
)
for
x
>
0
f(x)=\left\{\begin{array}{ll} x^{2} & \text { for } x \leq 0 \\ \ln (x) & \text { for } x>0 \end{array}\right.
f
(
x
)
=
{
x
2
ln
(
x
)
for
x
≤
0
for
x
>
0
\newline
Find
lim
x
→
1
f
(
x
)
\lim _{x \rightarrow 1} f(x)
lim
x
→
1
f
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
0
0
0
\newline
(B)
1
1
1
\newline
(C)
e
e
e
\newline
(D) The limit doesn't exist.
Get tutor help
h
(
x
)
=
{
x
2
−
1
for
x
≤
3
2
x
+
1
for
3
<
x
<
10
h(x)=\left\{\begin{array}{ll} x^{2}-1 & \text { for } x \leq 3 \\ 2 x+1 & \text { for } 3<x<10 \end{array}\right.
h
(
x
)
=
{
x
2
−
1
2
x
+
1
for
x
≤
3
for
3
<
x
<
10
\newline
Find
lim
x
→
3
−
h
(
x
)
\lim _{x \rightarrow 3^{-}} h(x)
lim
x
→
3
−
h
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
3
3
3
\newline
(B)
7
7
7
\newline
(C)
8
8
8
\newline
(D) The limit doesn't exist.
Get tutor help
Which of the following functions are continuous at
x
=
0
x=0
x
=
0
?
\newline
g
(
x
)
=
cot
(
x
)
g(x)=\cot (x)
g
(
x
)
=
cot
(
x
)
\newline
h
(
x
)
=
1
x
2
h(x)=\frac{1}{x^{2}}
h
(
x
)
=
x
2
1
\newline
Choose
1
1
1
answer:
\newline
(A)
g
g
g
only
\newline
(B)
h
h
h
only
\newline
(C) Both
g
g
g
and
h
h
h
\newline
(D) Neither
g
g
g
nor
h
h
h
Get tutor help
lim
x
→
0
tan
(
x
)
=
?
\lim _{x \rightarrow 0} \tan (x)=?
x
→
0
lim
tan
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
Get tutor help
lim
x
→
π
6
cot
(
x
)
=
?
\lim _{x \rightarrow \frac{\pi}{6}} \cot (x)=?
x
→
6
π
lim
cot
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
3
3
\frac{\sqrt{3}}{3}
3
3
\newline
(B)
3
2
\frac{\sqrt{3}}{2}
2
3
\newline
(C)
3
\sqrt{3}
3
\newline
(D) The limit doesn't exist.
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lim
x
→
π
cot
(
x
)
=
?
\lim _{x \rightarrow \pi} \cot (x)=?
x
→
π
lim
cot
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
Get tutor help
lim
x
→
π
2
sin
(
x
)
=
?
\lim _{x \rightarrow \frac{\pi}{2}} \sin (x)=?
x
→
2
π
lim
sin
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
Get tutor help
lim
x
→
0
cos
(
x
)
=
?
\lim _{x \rightarrow 0} \cos (x)=?
x
→
0
lim
cos
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
Get tutor help
lim
x
→
0
csc
(
x
)
=
?
\lim _{x \rightarrow 0} \csc (x)=?
x
→
0
lim
csc
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
Get tutor help
lim
x
→
π
cos
(
x
)
=
?
\lim _{x \rightarrow \pi} \cos (x)=?
x
→
π
lim
cos
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
Get tutor help
lim
x
→
π
tan
(
x
)
=
?
\lim _{x \rightarrow \pi} \tan (x)=?
x
→
π
lim
tan
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
1
1
1
\newline
(D) The limit doesn't exist.
Get tutor help
g
(
x
)
=
{
2
x
−
1
for
−
8
≤
x
<
1
x
for
x
≥
1
g(x)=\left\{\begin{array}{ll} 2^{x}-1 & \text { for }-8 \leq x<1 \\ \sqrt{x} & \text { for } x \geq 1 \end{array}\right.
g
(
x
)
=
{
2
x
−
1
x
for
−
8
≤
x
<
1
for
x
≥
1
\newline
Find
lim
x
→
4
g
(
x
)
\lim _{x \rightarrow 4} g(x)
lim
x
→
4
g
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
1
1
\newline
(B)
2
2
2
\newline
(C)
15
15
15
\newline
(D) The limit doesn't exist.
Get tutor help
h
(
x
)
=
{
5
x
for
x
<
−
2
x
3
−
2
for
x
≥
−
2
h(x)=\left\{\begin{array}{ll} 5 x & \text { for } x<-2 \\ x^{3}-2 & \text { for } x \geq-2 \end{array}\right.
h
(
x
)
=
{
5
x
x
3
−
2
for
x
<
−
2
for
x
≥
−
2
\newline
Find
lim
x
→
−
2
h
(
x
)
\lim _{x \rightarrow-2} h(x)
lim
x
→
−
2
h
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
10
-10
−
10
\newline
(B)
−
2
-2
−
2
\newline
(C)
10
10
10
\newline
(D) The limit doesn't exist.
Get tutor help
lim
x
→
π
4
cot
(
x
)
=
?
\lim _{x \rightarrow \frac{\pi}{4}} \cot (x)=?
x
→
4
π
lim
cot
(
x
)
=
?
\newline
Choose
1
1
1
answer:
\newline
(A)
2
2
\frac{\sqrt{2}}{2}
2
2
\newline
(B)
1
1
1
\newline
(C)
2
\sqrt{2}
2
\newline
(D) The limit doesn't exist.
Get tutor help
f
(
x
)
=
{
sin
(
x
)
for
0
≤
x
≤
π
x
π
−
1
for
π
<
x
≤
10
f(x)=\left\{\begin{array}{ll} \sin (x) & \text { for } 0 \leq x \leq \pi \\ \frac{x}{\pi}-1 & \text { for } \pi<x \leq 10 \end{array}\right.
f
(
x
)
=
{
sin
(
x
)
π
x
−
1
for
0
≤
x
≤
π
for
π
<
x
≤
10
\newline
Find
lim
x
→
π
f
(
x
)
\lim _{x \rightarrow \pi} f(x)
lim
x
→
π
f
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
\newline
(B)
0
0
0
\newline
(C)
π
\pi
π
\newline
(D) The limit doesn't exist.
Get tutor help
Find
lim
x
→
∞
4
x
2
−
3
x
x
3
\lim _{x \rightarrow \infty} \frac{4 x^{2}-3 x}{x^{3}}
lim
x
→
∞
x
3
4
x
2
−
3
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
4
4
4
\newline
(B)
0
0
0
\newline
(C)
−
3
-3
−
3
\newline
(D) The limit is unbounded
Get tutor help
Find
lim
x
→
∞
−
4
x
3
+
5
x
2
x
3
+
7
\lim _{x \rightarrow \infty} \frac{-4 x^{3}+5 x}{2 x^{3}+7}
lim
x
→
∞
2
x
3
+
7
−
4
x
3
+
5
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
5
7
\frac{5}{7}
7
5
\newline
(B)
−
2
-2
−
2
\newline
(C)
0
0
0
\newline
(D) The limit is unbounded
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Find
lim
x
→
∞
3
x
3
−
5
x
x
3
−
2
x
2
+
1
\lim _{x \rightarrow \infty} \frac{3 x^{3}-5 x}{x^{3}-2 x^{2}+1}
lim
x
→
∞
x
3
−
2
x
2
+
1
3
x
3
−
5
x
\newline
Choose
1
1
1
answer:
\newline
(A)
−
5
-5
−
5
\newline
(B)
0
0
0
\newline
(C)
3
3
3
\newline
(D) The limit is unbounded
Get tutor help
Find the derivative of
f
(
x
)
f(x)
f
(
x
)
.
\newline
f
(
x
)
=
ln
(
x
)
f(x) = \ln(x)
f
(
x
)
=
ln
(
x
)
\newline
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
______
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