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# Find the limit of $\left(1+3\ln\left(x\right)\right)^{\frac{1}{\sin\left(1-x\right)}}$ as $x$ approaches $1$

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##  Final answer to the problem

$\frac{1}{e^{3}}$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Solve using L'HÃ´pital's rule
• Solve without using l'HÃ´pital
• Solve using limit properties
• Solve using direct substitution
• Solve the limit using factorization
• Solve the limit using rationalization
• Integrate by partial fractions
• Product of Binomials with Common Term
• FOIL Method
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1

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

$\lim_{x\to1}\left(e^{\frac{1}{\sin\left(1-x\right)}\ln\left(1+3\ln\left(x\right)\right)}\right)$

Learn how to solve limits of exponential functions problems step by step online.

$\lim_{x\to1}\left(e^{\frac{1}{\sin\left(1-x\right)}\ln\left(1+3\ln\left(x\right)\right)}\right)$

Learn how to solve limits of exponential functions problems step by step online. Find the limit of (1+3ln(x))^(1/sin(1-x)) as x approaches 1. Rewrite the limit using the identity: a^x=e^{x\ln\left(a\right)}. Multiplying the fraction by \ln\left(1+3\ln\left(x\right)\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

$\frac{1}{e^{3}}$

$0.049787$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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0
a
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x
y
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(◻)
+
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×
◻/◻
/
÷
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

###  Main Topic: Limits of Exponential Functions

Those are limits of expressions of the form f(x)^g(x).