# How do you find the limit of (sin^4x)/(x^(1/2)) as x approaches infinity?

Aug 14, 2016

$0$

#### Explanation:

${\sin}^{4} x \in \left[- 1 , 1\right]$, $x \in m a t h c a l \left(R\right)$.

Because the numerator is continuous and bounded:

${\lim}_{x \to \infty} \frac{{\sin}^{4} x}{{x}^{\frac{1}{2}}}$

$= {\sin}^{4} x {\lim}_{x \to \infty} \frac{1}{{x}^{\frac{1}{2}}}$

by quotient rule
$= {\sin}^{4} x \frac{{\lim}_{x \to \infty} 1}{{\lim}_{x \to \infty} {x}^{\frac{1}{2}}}$

$= {\sin}^{4} x \cdot \frac{1}{\infty}$

$= 0$