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

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              2
              -
              x
 lim (1 + 2*x) 
x->0+

\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}}

Limit((1 + 2*x)^(2/x), x, 0)

Detail solution

Let's take the limit

\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}}

transform
do replacement

u = \frac{1}{2 x}

then

\lim_{x \to 0^+} \left(1 + \frac{2}{\frac{1}{x}}\right)^{\frac{2}{x}}

=
=

\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}

\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{4 u}

\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{4}

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)^{4} = e^{4}

The final answer:

\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}} = e^{4}

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]

              2
              -
              x
 lim (1 + 2*x) 
x->0+

\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}}

4
e

e^{4}

= 54.5981500331442

              2
              -
              x
 lim (1 + 2*x) 
x->0-

\lim_{x \to 0^-} \left(2 x + 1\right)^{\frac{2}{x}}

4
e

e^{4}

exp(4)

Other limits x→0, -oo, +oo, 1

\lim_{x \to 0^-} \left(2 x + 1\right)^{\frac{2}{x}} = e^{4}

\lim_{x \to 0^+} \left(2 x + 1\right)^{\frac{2}{x}} = e^{4}

\lim_{x \to \infty} \left(2 x + 1\right)^{\frac{2}{x}} = 1

\lim_{x \to 1^-} \left(2 x + 1\right)^{\frac{2}{x}} = 9

\lim_{x \to 1^+} \left(2 x + 1\right)^{\frac{2}{x}} = 9

\lim_{x \to -\infty} \left(2 x + 1\right)^{\frac{2}{x}} = 1

Rapid solution [src]

4
e

e^{4}

Expand and simplify

Numerical answer [src]

54.5981500331442

54.5981500331442

The graph