  Let $-\infty \leq a<b\leq \infty $. Let $f$ and $g$ be differentiable
functions on $(a,b)$ and let $g^{\prime }\neq 0$ on $(a,b)$. By introducing an
auxiliary function $H_{f,g}:=\left( f^{\prime }/g^{\prime }\right) g-f$, we
easily prove L'Hoipital rules for monotonicity. This offer a natural and
concise way so that those rules are easier to be understood. Using our
L'Hospital Piecewise Monotone Rules (for short, LPMR), we establish three new
sharp inequalities for hyperbolic and trigonometric functions as well as
bivariate means, which supplement certain known results.
