Question

Evaluate:
$\rm \displaystyle \lim_{x\to 0}\ \frac{\tan 5x}{\sin 3x}$

Answer 1

hope it is helpful...if so mark it as brilliant

Answer 2

As we know that

$\lim_{x\to 0}( \frac{tan \: x}{x} ) = 1 \\ \\ \lim_{x\to 0}( \frac{sin \: x}{x} ) = 1$
$\lim_{x\to 0}\ \frac{\tan 5x}{\sin 3x} \\ \\ \lim_{x\to 0} \frac{ (\frac{tan \: 5x}{5x} )5x}{( \frac{sin \: 3x}{3x})3x } \\ \\ \frac{5}{3} \frac{\lim_{x\to 0}(\frac{tan \: 5x}{5x} )}{\lim_{x\to 0}(\frac{sin \: 3x}{3x} )} \\ \\ = \frac{5}{3} \times\:\frac{1}{1} \\ \\ \lim_{x\to 0}\ \frac{\tan 5x}{\sin 3x} = \frac{5}{3}$
Hope it helps you.