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L'Hospital's Rule: Indeterminate Forms
Related Topics
Wize University Calculus 1 Textbook > Applications of Differentiation
Solving General Indeterminate Forms
4 Activities
Wize University Calculus 1 Textbook > Applications of Differentiation
L’Hopital’s Rule & Indeterminate Forms
3 Activities
Find
lim
x
→
0
+
(
ln
x
)
(
sin
x
)
\lim_{x\rightarrow0^+}\left(\ln x\right)\left(\sin x\right)
lim
x
→
0
+
(
ln
x
)
(
sin
x
)
.
a. 0
b. 1
c. 2
d.
e
e
e
e. None of the above
I don't know
Check Submission
More Solving General Indeterminate Forms Questions:
L'Hopital's Rule: General Indeterminate Forms
Evaluate the limit
L
=
lim
x
→
0
+
x
sin
x
L=\lim\limits_{x\rightarrow0^+}x^{\sin\ x}
L
=
x
→
0
+
lim
x
s
i
n
x
.
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
∞
(
1
−
k
x
)
3
x
\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}
lim
x
→
∞
(
1
−
x
k
)
3
x
.
Indeterminate Forms
Evaluate the limit if it exists.
lim
x
→
∞
(
1
+
1
x
)
x
\lim_{x\to\infty}\ \left(1+\frac{1}{x}\right)^x
lim
x
→
∞
(
1
+
x
1
)
x
.
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
∞
(
1
−
k
x
)
3
x
\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}
lim
x
→
∞
(
1
−
x
k
)
3
x
.
L'Hospital's Rule: Indeterminate Forms
Find
lim
x
→
0
+
(
ln
x
)
(
sin
x
)
\lim_{x\rightarrow0^+}\left(\ln x\right)\left(\sin x\right)
lim
x
→
0
+
(
ln
x
)
(
sin
x
)
.
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
0
+
(
e
1
x
)
1
ln
x
\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}
lim
x
→
0
+
(
e
x
1
)
l
n
x
1
.
Indeterminate Forms
Evaluate the limit:
lim
x
→
0
+
(
2
x
)
x
2
\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
x
→
0
+
lim
(
2
x
)
2
x
Indeterminate Forms: Polynomial and Ln
Evaluate the limit:
lim
x
→
0
+
x
a
ln
x
\displaystyle \lim_{x\rightarrow 0^+}x^a\ln{x}
x
→
0
+
lim
x
a
ln
x
, where
a
>
0
a>0
a
>
0
.
Practice: Fractions with Trig
Q.
\textbf{Q.}
Q.
Evaluate:
lim
x
→
0
+
(
2
x
−
2
sin
x
)
\displaystyle \lim_{x\rightarrow 0^+}\left(\frac{2}{x}-\frac{2}{\sin{x}}\right)
x
→
0
+
lim
(
x
2
−
sin
x
2
)
Evaluate the following limit:
lim
x
→
0
+
(
2
x
)
x
2
\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
x
→
0
+
lim
(
2
x
)
2
x
Evaluate the following limit:
lim
x
→
0
+
x
3
2
ln
x
\displaystyle \lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln{x}
x
→
0
+
lim
x
2
3
ln
x
Practice: $f^g$ Form with Trig
Q.
\textbf{Q.}
Q.
Evaluate the following limit:
lim
x
→
0
+
cos
x
cot
x
\displaystyle \lim_{x\rightarrow 0^+} \cos{x}^{\cot{x}}
x
→
0
+
lim
cos
x
c
o
t
x
Indeterminate Forms
Evaluate the limit:
lim
x
→
∞
[
1
+
sin
(
2
x
)
]
2
x
\displaystyle \lim_{x\rightarrow\infty}\left[1+\sin\left(\frac{2}{x}\right)\right]^{2x}
x
→
∞
lim
[
1
+
sin
(
x
2
)
]
2
x
Practice: L'Hospital's Rule
Evaluate
lim
x
→
0
+
(
e
1
x
)
1
ln
x
\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}
lim
x
→
0
+
(
e
x
1
)
l
n
x
1
.
Practice: L'Hospital's Rule
Evaluate
lim
x
→
∞
x
3
sin
(
1
x
2
)
\displaystyle \lim_{x\to\infty}\ x^3\ \sin\left(\frac{1}{x^2}\right)
x
→
∞
lim
x
3
sin
(
x
2
1
)
.
Practice: L'Hospital's Rule
Evaluate
lim
x
→
∞
x
3
sin
(
1
x
2
)
\displaystyle \lim_{x\to\infty}\ x^3\ \sin\left(\frac{1}{x^2}\right)
x
→
∞
lim
x
3
sin
(
x
2
1
)
.
General Indeterminate Forms
Evaluate:
lim
x
→
+
∞
2
x
+
1
9
x
2
−
2
x
−
6
\lim_{x \rightarrow +\infty} \frac{2x + 1}{\sqrt{9x^2-2x - 6}}
x
→
+
∞
lim
9
x
2
−
2
x
−
6
2
x
+
1
Practice: L'Hospital's Rule
Evaluate
lim
x
→
0
+
(
e
1
x
)
1
ln
x
\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}
lim
x
→
0
+
(
e
x
1
)
l
n
x
1
.
Practice: L'Hospital's Rule
Evaluate
lim
x
→
∞
x
3
sin
(
1
x
2
)
\displaystyle \lim_{x\to\infty}\ x^3\ \sin\left(\frac{1}{x^2}\right)
x
→
∞
lim
x
3
sin
(
x
2
1
)
.
Evaluate
lim
x
→
0
+
2
ln
x
ln
(
e
x
−
1
)
\lim_{x\rightarrow0^+}\frac{2\ln x}{\ln\left(e^x-1\right)}
lim
x
→
0
+
l
n
(
e
x
−
1
)
2
l
n
x
General Indeterminate Forms
Evaluate
lim
x
→
0
+
x
a
ln
x
where
a
>
0
\lim\limits_{x\to0^+} x^a\ln x \text{ where } a>0
x
→
0
+
lim
x
a
ln
x
where
a
>
0
l'Hopital's Rule: General Indeterminate Forms
Evaluate
lim
x
→
0
+
x
3
ln
x
\lim\limits_{x\to0^+}x^3\ln x
x
→
0
+
lim
x
3
ln
x
Evaluate the following limit:
lim
x
→
0
+
(
2
x
)
x
2
\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
x
→
0
+
lim
(
2
x
)
2
x
Evaluate the following limit:
lim
x
→
0
+
x
3
2
ln
x
\displaystyle \lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln{x}
x
→
0
+
lim
x
2
3
ln
x
Evaluate the following limit:
lim
x
→
0
+
x
3
2
ln
x
\displaystyle \lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln{x}
x
→
0
+
lim
x
2
3
ln
x
Evaluate the following limit:
lim
x
→
0
+
(
2
x
)
x
2
\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
x
→
0
+
lim
(
2
x
)
2
x
Indeterminate Forms: Power and Ln
Evaluate
lim
x
→
0
+
x
3
2
ln
x
\displaystyle \lim_{x\rightarrow 0^+} x^{\frac{3}{2}}\ln{x}
x
→
0
+
lim
x
2
3
ln
x
Practice: $f^g$ Form with Trig
Q.
\textbf{Q.}
Q.
Evaluate the following limit:
lim
x
→
0
+
cos
x
cot
x
\displaystyle \lim_{x\rightarrow 0^+} \cos{x}^{\cot{x}}
x
→
0
+
lim
cos
x
c
o
t
x
Practice: Fractions with Trig
Q.
\textbf{Q.}
Q.
Evaluate:
lim
x
→
0
+
(
2
x
−
2
sin
x
)
\displaystyle \lim_{x\rightarrow 0^+}\left(\frac{2}{x}-\frac{2}{\sin{x}}\right)
x
→
0
+
lim
(
x
2
−
sin
x
2
)
Indeterminate Forms
Evaluate the limit:
lim
x
→
∞
[
1
+
sin
(
2
x
)
]
2
x
\displaystyle \lim_{x\rightarrow\infty}\left[1+\sin\left(\frac{2}{x}\right)\right]^{2x}
x
→
∞
lim
[
1
+
sin
(
x
2
)
]
2
x
Indeterminate Forms: Polynomial and Ln
Evaluate the limit:
lim
x
→
0
+
x
a
ln
x
\displaystyle \lim_{x\rightarrow 0^+}x^a\ln{x}
x
→
0
+
lim
x
a
ln
x
, where
a
>
0
a>0
a
>
0
.
Indeterminate Forms
Evaluate the limit:
lim
x
→
0
+
(
2
x
)
x
2
\displaystyle \lim_{x\rightarrow 0^+}(2x)^{\frac{x}{2}}
x
→
0
+
lim
(
2
x
)
2
x
Determine
lim
x
→
∞
(
1
+
7
x
)
x
5
\lim_{x\rightarrow\infty}\left(1+\frac{7}{x}\right)^{\frac{x}{5}}
lim
x
→
∞
(
1
+
x
7
)
5
x
.
Determine
lim
x
→
∞
(
x
)
1
x
2
\lim\ _{x\rightarrow\infty}\left(\sqrt{x}\right)^{\frac{1}{x^2}}
lim
x
→
∞
(
x
)
x
2
1
.
Evaluate the limit
lim
x
→
∞
(
1
+
sin
2
x
)
2
x
\displaystyle \lim_{x\rightarrow \infty}\left(1+\sin{\frac{2}{x}}\right)^{2x}
x
→
∞
lim
(
1
+
sin
x
2
)
2
x
.
Evaluate the limit:
lim
x
→
0
+
(
2
x
−
2
sin
x
)
\displaystyle \lim_{x\rightarrow 0^+}\left(\frac{2}{x}-\frac{2}{\sin{x}}\right)
x
→
0
+
lim
(
x
2
−
sin
x
2
)
Solve the limit
lim
x
→
0
(
1
−
2
x
)
1
x
2
\displaystyle\lim_{x\ \to0}\left(1-2x\right)^{\frac{1}{x^2}}
x
→
0
lim
(
1
−
2
x
)
x
2
1
.
Find
lim
x
→
0
+
(
ln
x
)
(
sin
x
)
\lim_{x\rightarrow0^+}\left(\ln x\right)\left(\sin x\right)
lim
x
→
0
+
(
ln
x
)
(
sin
x
)
.
🦊
TRICKY!
Determine
lim
x
→
2
+
(
x
−
2
)
1
x
−
2
\lim_{x\rightarrow2^+}\left(x-2\right)^{\frac{1}{x-2}}
lim
x
→
2
+
(
x
−
2
)
x
−
2
1
.
L'Hopital's Rule: General Indeterminate Forms
Evaluate the limit
L
=
lim
x
→
0
+
x
sin
x
L=\lim\limits_{x\rightarrow0^+}x^{\sin\ x}
L
=
x
→
0
+
lim
x
s
i
n
x
.
l'Hopital's Rule: General Indeterminate Forms
Evaluate
lim
x
→
∞
(
1
+
7
x
)
x
/
5
\lim_{x\to \infty}\left(1+\frac{7}{x}\right)^{x/5}
x
→
∞
lim
(
1
+
x
7
)
x
/5
Evaluate
lim
x
→
0
+
x
3
2
ln
x
\lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln x
lim
x
→
0
+
x
2
3
ln
x
Determine
lim
x
→
∞
(
1
+
7
x
)
x
5
\lim_{x\rightarrow\infty}\left(1+\frac{7}{x}\right)^{\frac{x}{5}}
lim
x
→
∞
(
1
+
x
7
)
5
x
Indeterminate Forms
Evaluate the limit if it exists.
lim
x
→
∞
(
1
+
1
x
)
x
\lim_{x\to\infty}\ \left(1+\frac{1}{x}\right)^x
lim
x
→
∞
(
1
+
x
1
)
x
.
Indeterminate Forms
Evaluate the limit, if it exists.
lim
x
→
3
1
x
−
3
−
e
1
x
−
3
\lim_{x\to3}\ \frac{1}{x-3}-e^{\frac{1}{x-3}}
lim
x
→
3
x
−
3
1
−
e
x
−
3
1
.
Indeterminate Forms
Evaluate the limit, if it exists.
lim
x
→
−
2
(
x
+
2
)
ln
∣
x
+
2
∣
\lim_{x\to-2}\ \left(x+2\right)\ln\left|x+2\right|
lim
x
→
−
2
(
x
+
2
)
ln
∣
x
+
2
∣
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
∞
(
1
−
k
x
)
3
x
\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}
lim
x
→
∞
(
1
−
x
k
)
3
x
.
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
0
+
(
e
1
x
)
1
ln
x
\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}
lim
x
→
0
+
(
e
x
1
)
l
n
x
1
.
Evaluate
lim
x
→
0
+
x
3
2
ln
x
\lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln x
lim
x
→
0
+
x
2
3
ln
x
Practice: L'hopital's rule
Q.
\textbf{Q.}
Q.
Evaluate the following limits.
1.
lim
x
→
0
+
cos
x
cot
x
\displaystyle \lim_{x\rightarrow 0^+} \cos{x}^{\cot{x}}
x
→
0
+
lim
cos
x
c
o
t
x
2.
lim
x
→
∞
[
x
2
+
1
−
x
]
\displaystyle \lim_{x\rightarrow \infty}\left[\sqrt{x^2+1}-x\right]
x
→
∞
lim
[
x
2
+
1
−
x
]
More L’Hopital’s Rule & Indeterminate Forms Questions:
Evaluate the limit
L
=
lim
x
→
∞
ln
(
e
2
x
+
x
)
x
L=\lim\limits_{x\rightarrow \infin}\frac{\ln(e^{2x}+\ x)}{x}
L
=
x
→
∞
lim
x
l
n
(
e
2
x
+
x
)
.
L'Hopital's Rule: Trig and Ln
Find
lim
x
→
0
+
cot
x
ln
x
2
\displaystyle \lim_{x\rightarrow 0^+}\frac{\cot{x}}{\ln{x^2}}
x
→
0
+
lim
ln
x
2
cot
x
L'Hopital's Rule: One Sided Limits
Find
lim
x
→
0
e
x
−
1
1
−
cos
x
\lim_{x\to0}\frac{e^{x}-1}{1-\cos x}
lim
x
→
0
1
−
c
o
s
x
e
x
−
1
L'Hopital's Rule: Indeterminate Forms
Evaluate the following limit
lim
x
→
∞
x
2
+
x
e
x
+
1
\lim_{x\to\infty}\frac{x^{2}+x}{e^{x}+1}
lim
x
→
∞
e
x
+
1
x
2
+
x
L'Hopital's Rule and Indeterminate Forms
Evaluate
lim
x
→
0
+
(
2
x
−
2
sin
x
)
\lim\limits_{x\to0^+} \left(\dfrac{2}{x}-\dfrac{2}{\sin x}\right)
x
→
0
+
lim
(
x
2
−
sin
x
2
)
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
∞
(
1
−
k
x
)
3
x
\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}
lim
x
→
∞
(
1
−
x
k
)
3
x
.
Does the sequence
a
n
=
(
1
+
2
n
)
n
a_n=\left(1+\frac{2}{n}\right)^n
a
n
=
(
1
+
n
2
)
n
converge or diverge?
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
∞
(
1
−
k
x
)
3
x
\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}
lim
x
→
∞
(
1
−
x
k
)
3
x
.
L'Hospital's Rule: Indeterminate Forms
Find
lim
x
→
0
+
(
ln
x
)
(
sin
x
)
\lim_{x\rightarrow0^+}\left(\ln x\right)\left(\sin x\right)
lim
x
→
0
+
(
ln
x
)
(
sin
x
)
.
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
0
+
(
e
1
x
)
1
ln
x
\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}
lim
x
→
0
+
(
e
x
1
)
l
n
x
1
.
L'Hospital's Rule: Indeterminate Forms
Determine
lim
x
→
0
tan
x
e
x
−
1
\lim_{x\rightarrow0}\frac{\tan x}{e^x-1}
lim
x
→
0
e
x
−
1
t
a
n
x
.
L'Hopital's Rule: Exponential and Ln's
Evaluate
lim
x
→
0
+
2
ln
x
ln
(
e
x
−
1
)
\displaystyle \lim_{x\rightarrow 0^+} \frac{2\ln{x}}{\ln(e^x-1)}
x
→
0
+
lim
ln
(
e
x
−
1
)
2
ln
x
L'Hopital's Rule: Trig and Ln
Find
lim
x
→
0
+
cot
x
ln
x
2
\displaystyle \lim_{x\rightarrow 0^+}\frac{\cot{x}}{\ln{x^2}}
x
→
0
+
lim
ln
x
2
cot
x
L'Hopital's Rule: One Sided Limits
Q.
\textbf{Q.}
Q.
Find
lim
x
→
0
e
x
−
1
1
−
cos
5
x
\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{1-\cos{5x}}
x
→
0
lim
1
−
cos
5
x
e
x
−
1
Practice: L'Hospital's Rule
Determine
lim
x
→
0
tan
x
e
x
−
1
\displaystyle \lim_{x\rightarrow0}\frac{\tan x}{e^x-1}
x
→
0
lim
e
x
−
1
tan
x
.
Practice: L'Hospital's Rule
Determine
lim
x
→
0
tan
x
e
x
−
1
\displaystyle \lim_{x\rightarrow0}\frac{\tan x}{e^x-1}
x
→
0
lim
e
x
−
1
tan
x
.
L'Hopital's Rule: Indeterminate Forms
Compute the limit
lim
x
→
0
+
1
−
cos
2
x
x
3
\displaystyle \lim_{x \rightarrow 0^+} \frac{1 - \cos^2x}{x^3}
x
→
0
+
lim
x
3
1
−
cos
2
x
L'Hopital's Rule
Evaluate the limit (if it exists)
lim
x
→
0
(
e
1
x
)
tan
x
\displaystyle \lim_{x\to0}\ \left(e^{\frac{1}{x}}\right)^{\tan x}
x
→
0
lim
(
e
x
1
)
t
a
n
x
L'Hopital's Rule: Indeterminate Forms
The limit
lim
x
→
1
x
2
−
2
+
sin
(
π
x
2
)
ln
x
\displaystyle\lim_{x\to1}\ \frac{x^2-2+\sin\left(\frac{\pi x}{2}\right)}{\ln x}
x
→
1
lim
ln
x
x
2
−
2
+
sin
(
2
π
x
)
is
Practice: L'Hospital's Rule
Evaluate
lim
x
→
∞
(
1
−
1
x
)
3
x
\displaystyle \lim_{x\rightarrow\infty}\left(1-\frac{1}{x}\right)^{3x}
x
→
∞
lim
(
1
−
x
1
)
3
x
.
L'Hopital's Rule: Indeterminate Forms
lim
x
→
∞
x
2
⋅
sin
(
1
x
)
=
\displaystyle\lim_{x\to\infty}\ x^2\cdot\sin\left(\frac{1}{x}\right)=
x
→
∞
lim
x
2
⋅
sin
(
x
1
)
=
Practice: L'Hospital's Rule
Determine
lim
x
→
0
tan
x
e
x
−
1
\displaystyle \lim_{x\rightarrow0}\frac{\tan x}{e^x-1}
x
→
0
lim
e
x
−
1
tan
x
.
Evaluate
lim
x
→
0
+
x
3
2
ln
x
\lim_{x\rightarrow0^+}x^{\frac{3}{2}}\ln x
lim
x
→
0
+
x
2
3
ln
x
Determine
lim
x
→
∞
(
1
+
7
x
)
x
5
\lim_{x\rightarrow\infty}\left(1+\frac{7}{x}\right)^{\frac{x}{5}}
lim
x
→
∞
(
1
+
x
7
)
5
x
L'Hopital's Rule: Indeterminate Forms
Does the sequence
a
n
=
(
1
+
2
n
)
n
a_n=\left(1+\frac{2}{n}\right)^n
a
n
=
(
1
+
n
2
)
n
converge or diverge?
Does the sequence
a
n
=
(
1
+
2
n
)
n
a_n=\left(1+\frac{2}{n}\right)^n
a
n
=
(
1
+
n
2
)
n
converge or diverge?
Does the sequence
a
n
=
(
1
+
2
n
)
n
a_n=\left(1+\frac{2}{n}\right)^n
a
n
=
(
1
+
n
2
)
n
converge or diverge?
Does the sequence
a
n
=
(
1
+
2
n
)
n
a_n=\left(1+\frac{2}{n}\right)^n
a
n
=
(
1
+
n
2
)
n
converge or diverge?
Calculate
lim
x
→
0
1
+
2
x
−
e
2
x
x
2
\displaystyle \lim_{x\rightarrow0}\frac{1+2x-e^{2x}}{x^2}
x
→
0
lim
x
2
1
+
2
x
−
e
2
x
L'Hopital's Rule and Indeterminate Forms
Evaluate
lim
x
→
0
+
(
2
x
−
2
sin
x
)
\lim\limits_{x\to0^+} \left(\dfrac{2}{x}-\dfrac{2}{\sin x}\right)
x
→
0
+
lim
(
x
2
−
sin
x
2
)
L'Hopital's Rule" Indeterminate Forms
Evaluate
lim
x
→
−
1
x
4
+
2
x
+
1
x
3
−
2
x
−
1
\lim\limits_{x \to -1} \dfrac{x^4+2x+1}{x^3-2x-1}
x
→
−
1
lim
x
3
−
2
x
−
1
x
4
+
2
x
+
1
l'Hopital's Rule: Indeterminate Forms
Evaluate
lim
x
→
∞
x
2
+
x
e
2
x
+
1
\lim\limits_{x\to\infty}\dfrac{x^2+x}{e^{2x}+1}
x
→
∞
lim
e
2
x
+
1
x
2
+
x
L'Hopital's Rule: Indeterminate Forms
Evaluate
lim
x
→
∞
(
1
−
k
x
)
3
x
\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}
lim
x
→
∞
(
1
−
x
k
)
3
x
.
Evaluate the limit:
lim
x
→
0
2
sin
x
−
sin
(
2
x
)
2
e
x
−
2
−
x
2
\displaystyle \lim_{x\rightarrow 0}\frac{2\sin{x}-\sin{(2x)}}{2e^x-2-x^2}
x
→
0
lim
2
e
x
−
2
−
x
2
2
sin
x
−
sin
(
2
x
)
Evaluate the following limit
lim
x
→
∞
x
2
+
x
e
2
x
+
1
\displaystyle \lim_{x\rightarrow\infty}\frac{x^2+x}{e^{2x}+1}
x
→
∞
lim
e
2
x
+
1
x
2
+
x
.
Does the sequence
a
n
=
(
1
+
2
n
)
n
a_n=\left(1+\frac{2}{n}\right)^n
a
n
=
(
1
+
n
2
)
n
converge or diverge?
Evaluate the limit:
lim
x
→
0
2
sin
x
−
sin
(
2
x
)
2
e
x
−
2
−
x
2
\displaystyle \lim_{x\rightarrow 0}\frac{2\sin{x}-\sin{(2x)}}{2e^x-2-x^2}
x
→
0
lim
2
e
x
−
2
−
x
2
2
sin
x
−
sin
(
2
x
)
L'Hopital's Rule: One Sided Limits
Q.
\textbf{Q.}
Q.
Find
lim
x
→
0
e
x
−
1
1
−
cos
5
x
\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{1-\cos{5x}}
x
→
0
lim
1
−
cos
5
x
e
x
−
1
L'Hopital's Rule: Trig and Ln
Find
lim
x
→
0
+
cot
x
ln
x
2
\displaystyle \lim_{x\rightarrow 0^+}\frac{\cot{x}}{\ln{x^2}}
x
→
0
+
lim
ln
x
2
cot
x
L'Hopital's Rule: Exponential and Ln's
Evaluate
lim
x
→
0
+
2
ln
x
ln
(
e
x
−
1
)
\displaystyle \lim_{x\rightarrow 0^+} \frac{2\ln{x}}{\ln(e^x-1)}
x
→
0
+
lim
ln
(
e
x
−
1
)
2
ln
x
🌶️
TOUGH!
Evaluate the limit
L
=
lim
x
→
∞
ln
(
e
2
x
+
x
)
x
L=\lim\limits_{x\rightarrow \infin}\frac{\ln(e^{2x}+\ x)}{x}
L
=
x
→
∞
lim
x
l
n
(
e
2
x
+
x
)
.
L'Hopital's Rule: Indeterminate Forms
Evaluate the following limit
lim
x
→
∞
x
2
+
x
e
2
x
+
1
\displaystyle \lim_{x\rightarrow\infty}\frac{x^2+x}{e^{2x}+1}
x
→
∞
lim
e
2
x
+
1
x
2
+
x
Find the numerical value of the limit
L
=
lim
x
→
∞
sin
(
2
/
x
)
sin
(
1
/
x
)
L=\lim\limits_{x\rightarrow\infin}\frac{\sin(2/x)}{\sin(1/x)}
L
=
x
→
∞
lim
s
i
n
(
1/
x
)
s
i
n
(
2/
x
)
.
Evaluate the limit
lim
x
→
0
e
−
4
x
−
cos
(
3
x
)
5
x
\displaystyle \lim_{x\rightarrow 0}\frac{e^{-4x}-\cos{(3x)}}{5x}
x
→
0
lim
5
x
e
−
4
x
−
cos
(
3
x
)
.
🦊
TRICKY!
Evaluate
lim
x
→
0
sin
2
x
−
3
sin
x
sin
x
\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}-3\sin{x}}{\sin{x}}
x
→
0
lim
sin
x
sin
2
x
−
3
sin
x
. Hint: use trigonometric identity
sin
(
2
x
)
=
2
sin
x
cos
x
\sin(2x)=2\sin x\cos x
sin
(
2
x
)
=
2
sin
x
cos
x
.
Evaluate
lim
x
→
0
+
2
ln
x
ln
(
e
x
−
1
)
\lim_{x\rightarrow0^+}\frac{2\ln x}{\ln\left(e^x-1\right)}
lim
x
→
0
+
l
n
(
e
x
−
1
)
2
l
n
x
.
Evaluate the following limit
lim
x
→
0
e
x
−
1
1
−
cos
5
x
\displaystyle \lim_{x\rightarrow 0}\frac{e^x-1}{1-\cos{5x}}
x
→
0
lim
1
−
cos
5
x
e
x
−
1
.
Evaluate the limit
lim
x
→
0
2
sin
x
−
sin
(
2
x
)
2
e
x
−
2
−
x
2
\displaystyle \lim_{x\rightarrow 0}\frac{2\sin{x}-\sin{(2x)}}{2e^x-2-x^2}
x
→
0
lim
2
e
x
−
2
−
x
2
2
sin
x
−
sin
(
2
x
)
.
Calculate
lim
x
→
0
1
+
2
x
−
e
2
x
x
2
\displaystyle \lim_{x\rightarrow0}\frac{1+2x-e^{2x}}{x^2}
x
→
0
lim
x
2
1
+
2
x
−
e
2
x
Evaluate
lim
x
→
0
+
2
ln
x
ln
(
e
x
−
1
)
\lim_{x\rightarrow0^+}\frac{2\ln x}{\ln\left(e^x-1\right)}
lim
x
→
0
+
l
n
(
e
x
−
1
)
2
l
n
x
(Duplicated)
If
lim
x
→
0
x
f
(
x
)
=
a
\lim_{x\to0}\ \frac{x}{f\left(x\right)}=a
lim
x
→
0
f
(
x
)
x
=
a
, where
a
a
a
is a non-zero constant, find
lim
x
→
0
f
′
(
x
)
\lim_{x\to0}\ f'\left(x\right)
lim
x
→
0
f
′
(
x
)
.
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
∞
(
1
−
k
x
)
3
x
\lim\ _{x\rightarrow\infty}\left(1-\frac{k}{x}\right)^{3x}
lim
x
→
∞
(
1
−
x
k
)
3
x
.
L'Hospital's Rule: Indeterminate Forms
Evaluate
lim
x
→
0
+
(
e
1
x
)
1
ln
x
\lim_{x\rightarrow0^+}\left(e^{\frac{1}{x}}\right)^{^{\frac{1}{\ln x}}}
lim
x
→
0
+
(
e
x
1
)
l
n
x
1
.
Calculate
lim
x
→
0
1
+
2
x
−
e
2
x
x
2
\displaystyle \lim_{x\rightarrow0}\frac{1+2x-e^{2x}}{x^2}
x
→
0
lim
x
2
1
+
2
x
−
e
2
x
Practice: L'hopital's rule
Q.
\textbf{Q.}
Q.
Evaluate the following limits.
1.
lim
x
→
0
+
cos
x
cot
x
\displaystyle \lim_{x\rightarrow 0^+} \cos{x}^{\cot{x}}
x
→
0
+
lim
cos
x
c
o
t
x
2.
lim
x
→
∞
[
x
2
+
1
−
x
]
\displaystyle \lim_{x\rightarrow \infty}\left[\sqrt{x^2+1}-x\right]
x
→
∞
lim
[
x
2
+
1
−
x
]