What is the limit as x approaches 0+of (((1+x)^{1/x})/e)^{1/x} ?

The limit as x approaches 0+of (((1+x)^{1/x})/e)^{1/x} is 1/(sqrt(e))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

limit as x approaches 0+of (((1+x)^{1/x})/e)^{1/x}

x \to 0 + lim (\frac{( 1 + x ) ^{\frac{1}{x}}}{e})^{\frac{1}{x}}

\frac{1}{e}

Decimal

0.60653 \dots

Solution steps

x \to 0 + lim (\frac{( 1 + x ) ^{\frac{1}{x}}}{e})^{\frac{1}{x}}

Apply exponent rules

= x \to 0 + lim e^{\frac{1}{x} l n (\frac{( 1 + x ) ^{\frac{1}{x}}}{e})}

Apply the Limit Chain Rule:

\frac{1}{e}

= \frac{1}{e}

What is the limit as x approaches 0+of (((1+x)^{1/x})/e)^{1/x} ?
The limit as x approaches 0+of (((1+x)^{1/x})/e)^{1/x} is 1/(sqrt(e))