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(1-2*x)^(1/x)
Limit of the function/ (1-2*x)^(1/x)

Limit of the function (1-2*x)^(1/x)

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     x _________
 lim \/ 1 - 2*x 
x->0+
lim⁡x→0+(1−2x)1x\lim_{x \to 0^+} \left(1 - 2 x\right)^{\frac{1}{x}} 
Limit((1 - 2*x)^(1/x), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(1−2x)1x\lim_{x \to 0^+} \left(1 - 2 x\right)^{\frac{1}{x}}
transform
do replacement
u=1(−2)xu = \frac{1}{\left(-2\right) x}
then
lim⁡x→0+(1−21x)1x\lim_{x \to 0^+} \left(1 - \frac{2}{\frac{1}{x}}\right)^{\frac{1}{x}} =
=
lim⁡u→0+(1+1u)−2u\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 2 u}
=
lim⁡u→0+(1+1u)−2u\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 2 u}
=
((lim⁡u→0+(1+1u)u))−2\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-2}
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))−2=e−2\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-2} = e^{-2}

The final answer:
lim⁡x→0+(1−2x)1x=e−2\lim_{x \to 0^+} \left(1 - 2 x\right)^{\frac{1}{x}} = e^{-2}
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
02468-8-6-4-2-10100.01.0
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(1−2x)1x=e−2\lim_{x \to 0^-} \left(1 - 2 x\right)^{\frac{1}{x}} = e^{-2}
More at x→0 from the left
lim⁡x→0+(1−2x)1x=e−2\lim_{x \to 0^+} \left(1 - 2 x\right)^{\frac{1}{x}} = e^{-2}
lim⁡x→∞(1−2x)1x=1\lim_{x \to \infty} \left(1 - 2 x\right)^{\frac{1}{x}} = 1
More at x→oo
lim⁡x→1−(1−2x)1x=−1\lim_{x \to 1^-} \left(1 - 2 x\right)^{\frac{1}{x}} = -1
More at x→1 from the left
lim⁡x→1+(1−2x)1x=−1\lim_{x \to 1^+} \left(1 - 2 x\right)^{\frac{1}{x}} = -1
More at x→1 from the right
lim⁡x→−∞(1−2x)1x=1\lim_{x \to -\infty} \left(1 - 2 x\right)^{\frac{1}{x}} = 1
More at x→-oo
One‐sided limits [src] 
     x _________
 lim \/ 1 - 2*x 
x->0+
lim⁡x→0+(1−2x)1x\lim_{x \to 0^+} \left(1 - 2 x\right)^{\frac{1}{x}} 
 -2
e
e−2e^{-2} 
= 0.135335283236613
     x _________
 lim \/ 1 - 2*x 
x->0-
lim⁡x→0−(1−2x)1x\lim_{x \to 0^-} \left(1 - 2 x\right)^{\frac{1}{x}} 
 -2
e
e−2e^{-2} 
= 0.135335283236613
= 0.135335283236613
Rapid solution [src] 
 -2
e
e−2e^{-2}
Expand and simplify
Numerical answer [src] 
0.135335283236613
0.135335283236613
The graph
Limit of the function (1-2*x)^(1/x)
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