What is the limit as x approaches infinity+of (1+2x)^{1/(2ln(x))} ?

The limit as x approaches infinity+of (1+2x)^{1/(2ln(x))} is sqrt(e)

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limit as x approaches infinity+of (1+2x)^{1/(2ln(x))}

x \to \infty + lim ((1 + 2 x)^{\frac{1}{2 l n ( x )}})

e

Decimal

1.64872 \dots

Solution steps

x \to \infty lim ((1 + 2 x)^{\frac{1}{2 l n ( x )}})

Apply exponent rules

= x \to \infty lim (e^{\frac{1}{2 l n ( x )} l n (1 + 2 x)})

Apply the Limit Chain Rule:

e

= e

\int \frac{e ^{a r c t a n (y)}}{1 + y ^{2}} d y

\int (e^{2 x} cos (3 x)) d x

\int \frac{cos ( x )}{1 + 2 sin ( x )} d x

\frac{d}{d x} (e^{l n (x^{2})})

\frac{d}{d x} (\frac{x}{( 1 - x ) ^{2}})

What is the limit as x approaches infinity+of (1+2x)^{1/(2ln(x))} ?
The limit as x approaches infinity+of (1+2x)^{1/(2ln(x))} is sqrt(e)