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Limit of the function (1+3*x)^(5/x)

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              5
              -
              x
 lim (1 + 3*x) 
x->0+

\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}}

Limit((1 + 3*x)^(5/x), x, 0)

Detail solution

Let's take the limit

\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}}

transform
do replacement

u = \frac{1}{3 x}

then

\lim_{x \to 0^+} \left(1 + \frac{3}{\frac{1}{x}}\right)^{\frac{5}{x}}

=
=

\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{15 u}

\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{15 u}

\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{15}

The limit

\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}

is second remarkable limit, is equal to e ~ 2.718281828459045
then

\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{15} = e^{15}

The final answer:

\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}} = e^{15}

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

Plot the graph

One‐sided limits [src]

              5
              -
              x
 lim (1 + 3*x) 
x->0+

\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}}

 15
e

e^{15}

= 3269017.37247211

              5
              -
              x
 lim (1 + 3*x) 
x->0-

\lim_{x \to 0^-} \left(3 x + 1\right)^{\frac{5}{x}}

 15
e

e^{15}

exp(15)

Rapid solution [src]

 15
e

e^{15}

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to 0^-} \left(3 x + 1\right)^{\frac{5}{x}} = e^{15}

\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}} = e^{15}

\lim_{x \to \infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1

\lim_{x \to 1^-} \left(3 x + 1\right)^{\frac{5}{x}} = 1024

\lim_{x \to 1^+} \left(3 x + 1\right)^{\frac{5}{x}} = 1024

\lim_{x \to -\infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1

Numerical answer [src]

3269017.37247211

3269017.37247211

The graph