Question

evaluate. lim.x tends to 0 sinx - tanx/ x3

Answer 1

Solution:
$lim \: x \: -> 0 \: \: \frac{sinx - tan \: x}{ {x}^{3} }$
We know that

$\tan(x) = \frac{sin \: x}{cos \: x}$
$lim \: x \: -> 0 \: \: \frac{sinx - \frac{sin \: x}{cos \: x} }{ {x}^{3} } \\ \\lim \: x \: -> 0 \: \frac{sin \: x \: cos \: x - sin \: x}{cos \: x \: {x}^{3} } \\ \\lim \: x \: -> 0 \: \: \frac{sin \: x(cos \: x - 1)}{cos \: x \: {x}^{3} } \\ \\lim \: x \: -> 0 \: \: \frac{tan \: x}{x} \: \: lim \: x \: -> 0 \frac{ - 2 {sin}^{2} \frac{x}{2} }{ {x}^{2} } \\ \\ = > 1. \: \: lim \: x \: -> 0 \frac{ - \frac{2}{4} {sin}^{2} \frac{x}{2} }{ {( \frac{x}{2}) }^{2} } \\ \\ = > 1.( \frac{ - 1}{2} ) \\ \\ = - \frac{1}{2}$
Hope it helps you.