Question

Evaluate : $\lim_{x \to 0} \left[ 1 + \frac{5x}{6} \right]^{\frac{1}{x}}$

Answer 1

$e^{5/6}$

Answer 2

The limit is of the form $\lim_{x \to 0} \left[ 1 + \frac{5x}{6} \right]^{\frac{1}{x}}$.

This is an indeterminate form of type $1^\infty$.

Using the standard limit formula $\lim_{x \to 0} (1 + ax)^{1/x} = e^a$, we compare the given limit with this form.

Here, $a = \frac{5}{6}$.

Therefore, the value of the limit is $e^{5/6}$.