Question

Evaluate:
$\rm \displaystyle \lim_{x \to 0}\ \bigg(1-\frac{3x}{5}\bigg)^{\frac{1}{x}}$

Answer 1

Since replacing x with 0 results in 0/0, We use the L'Hopital's rule.

$\lim_{ x \to 0} \bigg( 1 - \frac{3x}{5} \bigg) ^{ \frac{1}{x} } \\ = e^{\lim_{ x \to 0} \frac{1}{x} \text{ ln} \bigg(1 - \frac{3x}{5} \bigg) } \\ = e^{\lim_{ x \to 0} \frac{ \frac{d}{dx} \text{ ln} (1 - \frac{3x}{5} )}{ \frac{d}{dx} x} } \\ = e^{\lim_{ x \to 0} \frac{3}{3x - 5} } \\ = e ^{ - \frac{ 3}{5} } \\ = \frac{1}{ {e}^{ \frac{3}{5} } }$

Answer 2

❤see in the pic it is been solved