Koundinya-Mathematics Teachers Hub: One more for you to try

For
${\color{Yellow} 0\leq t\leq 1}$
Show that

${\color{blue} t\leq tant\leq t+t^{3}}$

Now, for
${\color{blue} 0\leq x\leq \Pi}$
and
${\color{Yellow} 0\leq y\leq 1}$
Show that

${\color{blue} ysinx\leq tan(ysinx)\leq ysinx+(ysinx)^{3}}$

Finally, show that the limit
${\color{Yellow} L = \lim_{y\rightarrow 0}\frac{1}{y}\int_{0}^{\Pi }tan(ysinx)dx = 2}$