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Find the limit of $\left(1-e^x\right)^{\frac{1}{\ln\left(x\right)}}$ as $x$ approaches 0

Step-by-step Solution

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Final answer to the problem

$e$
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Step-by-step Solution

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  • Integrate by partial fractions
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Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$

$\lim_{x\to0}\left(e^{\frac{1}{\ln\left(x\right)}\ln\left(1-e^x\right)}\right)$

Learn how to solve differential calculus problems step by step online.

$\lim_{x\to0}\left(e^{\frac{1}{\ln\left(x\right)}\ln\left(1-e^x\right)}\right)$

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Learn how to solve differential calculus problems step by step online. Find the limit of (1-e^x)^(1/ln(x)) as x approaches 0. Rewrite the limit using the identity: a^x=e^{x\ln\left(a\right)}. Multiplying the fraction by \ln\left(1-e^x\right). Apply the power rule of limits: \displaystyle{\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\displaystyle\lim_{x\to a}g(x)}}. The limit of a constant is just the constant.

Final answer to the problem

$e$

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Function Plot

Plotting: $\left(1-e^x\right)^{\frac{1}{\ln\left(x\right)}}$

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5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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