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# Find the limit of $x^{\frac{1}{x}}$ as $x$ approaches $\infty$

## Step-by-step Solution

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##  Step-by-step Solution 

Problem to solve:

$\lim_{x\to\infty }\left(x^{\frac{1}{x}}\right)$

Specify the solving method

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

$\lim_{x\to\infty }\left(e^{\frac{1}{x}\ln\left(x\right)}\right)$

Learn how to solve limits to infinity problems step by step online.

$\lim_{x\to\infty }\left(e^{\frac{1}{x}\ln\left(x\right)}\right)$

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

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##  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

Limits by Direct SubstitutionLimits by L'Hôpital's ruleLimits by factoringLimits by rationalizing

SnapXam A2

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a
b
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f
g
m
n
u
v
w
x
y
z
.
(◻)
+
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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 to Infinity

~ 0.07 s

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