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(1+x)^(1/x)
Limit of the function/ (1+x)^(1/x)

Limit of the function (1+x)^(1/x)

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     x _______
 lim \/ 1 + x 
x->0+
lim⁡x→0+(x+1)1x\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}} 
Limit((1 + x)^(1/x), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(x+1)1x\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}}
transform
do replacement
u=1xu = \frac{1}{x}
then
lim⁡x→0+(1+11x)1x\lim_{x \to 0^+} \left(1 + \frac{1}{\frac{1}{x}}\right)^{\frac{1}{x}} =
=
lim⁡u→0+(1+1u)u\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}
=
lim⁡u→0+(1+1u)u\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}
=
((lim⁡u→0+(1+1u)u))\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)
The limit
lim⁡u→0+(1+1u)u\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}
is second remarkable limit, is equal to e ~ 2.718281828459045
then
((lim⁡u→0+(1+1u)u))=e\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right) = e

The final answer:
lim⁡x→0+(x+1)1x=e\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}} = e
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
02468-8-6-4-2-1010020
Plot the graph
One‐sided limits [src] 
     x _______
 lim \/ 1 + x 
x->0+
lim⁡x→0+(x+1)1x\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}} 
E
ee 
= 2.71828182845905
     x _______
 lim \/ 1 + x 
x->0-
lim⁡x→0−(x+1)1x\lim_{x \to 0^-} \left(x + 1\right)^{\frac{1}{x}} 
E
ee 
= 2.71828182845905
= 2.71828182845905
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(x+1)1x=e\lim_{x \to 0^-} \left(x + 1\right)^{\frac{1}{x}} = e
More at x→0 from the left
lim⁡x→0+(x+1)1x=e\lim_{x \to 0^+} \left(x + 1\right)^{\frac{1}{x}} = e
lim⁡x→∞(x+1)1x=1\lim_{x \to \infty} \left(x + 1\right)^{\frac{1}{x}} = 1
More at x→oo
lim⁡x→1−(x+1)1x=2\lim_{x \to 1^-} \left(x + 1\right)^{\frac{1}{x}} = 2
More at x→1 from the left
lim⁡x→1+(x+1)1x=2\lim_{x \to 1^+} \left(x + 1\right)^{\frac{1}{x}} = 2
More at x→1 from the right
lim⁡x→−∞(x+1)1x=1\lim_{x \to -\infty} \left(x + 1\right)^{\frac{1}{x}} = 1
More at x→-oo
Rapid solution [src] 
E
ee
Expand and simplify
Numerical answer [src] 
2.71828182845905
2.71828182845905
The graph
Limit of the function (1+x)^(1/x)
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