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(1-3*x)^(1/x)
Limit of the function/ (1-3*x)^(1/x)

Limit of the function (1-3*x)^(1/x)

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     x _________
 lim \/ 1 - 3*x 
x->0+
lim⁡x→0+(1−3x)1x\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}} 
Limit((1 - 3*x)^(1/x), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(1−3x)1x\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}}
transform
do replacement
u=1(−3)xu = \frac{1}{\left(-3\right) x}
then
lim⁡x→0+(1−31x)1x\lim_{x \to 0^+} \left(1 - \frac{3}{\frac{1}{x}}\right)^{\frac{1}{x}} =
=
lim⁡u→0+(1+1u)−3u\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 3 u}
=
lim⁡u→0+(1+1u)−3u\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- 3 u}
=
((lim⁡u→0+(1+1u)u))−3\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-3}
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))−3=e−3\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{-3} = e^{-3}

The final answer:
lim⁡x→0+(1−3x)1x=e−3\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}} = e^{-3}
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
Rapid solution [src] 
 -3
e
e−3e^{-3}
Expand and simplify
One‐sided limits [src] 
     x _________
 lim \/ 1 - 3*x 
x->0+
lim⁡x→0+(1−3x)1x\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}} 
 -3
e
e−3e^{-3} 
= 0.0497870683678639
     x _________
 lim \/ 1 - 3*x 
x->0-
lim⁡x→0−(1−3x)1x\lim_{x \to 0^-} \left(1 - 3 x\right)^{\frac{1}{x}} 
 -3
e
e−3e^{-3} 
= 0.0497870683678639
= 0.0497870683678639
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(1−3x)1x=e−3\lim_{x \to 0^-} \left(1 - 3 x\right)^{\frac{1}{x}} = e^{-3}
More at x→0 from the left
lim⁡x→0+(1−3x)1x=e−3\lim_{x \to 0^+} \left(1 - 3 x\right)^{\frac{1}{x}} = e^{-3}
lim⁡x→∞(1−3x)1x=1\lim_{x \to \infty} \left(1 - 3 x\right)^{\frac{1}{x}} = 1
More at x→oo
lim⁡x→1−(1−3x)1x=−2\lim_{x \to 1^-} \left(1 - 3 x\right)^{\frac{1}{x}} = -2
More at x→1 from the left
lim⁡x→1+(1−3x)1x=−2\lim_{x \to 1^+} \left(1 - 3 x\right)^{\frac{1}{x}} = -2
More at x→1 from the right
lim⁡x→−∞(1−3x)1x=1\lim_{x \to -\infty} \left(1 - 3 x\right)^{\frac{1}{x}} = 1
More at x→-oo
Numerical answer [src] 
0.0497870683678639
0.0497870683678639
The graph
Limit of the function (1-3*x)^(1/x)
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