# How do you find the limit of (xcos(ax))/(tan(bx)) as x approaches 0?

##### 1 Answer
May 10, 2016

Limit of $\frac{x \cos \left(a x\right)}{\tan} \left(b x\right)$, as $x$ approaches $0$ is $\frac{1}{b}$

#### Explanation:

$\frac{x \cos \left(a x\right)}{\tan} \left(b x\right) = \frac{x \cos \left(a x\right)}{\sin \frac{b x}{\cos} \left(b x\right)}$

or $\frac{x \cos \left(a x\right) \cos \left(b x\right)}{\sin} \left(b x\right)$

Hence ${L}_{x \to 0} \frac{x \cos \left(a x\right)}{\tan} \left(b x\right) = {L}_{x \to 0} \frac{x \cos \left(a x\right) \cos \left(b x\right)}{\sin} \left(b x\right)$

or ${L}_{x \to 0} \frac{1}{b} \times \frac{b x}{\sin} \left(b x\right) \times {L}_{x \to 0} \cos \left(a x\right) \times {L}_{x \to 0} \cos \left(b x\right)$

but as $x \to 0$, $\cos p x \to 1$ and ${L}_{z \to 0} \frac{z}{\sin} z = 1$,

the above is equal to

$\frac{1}{b} \times {L}_{x \to 0} \frac{b x}{\sin} \left(b x\right) \times {L}_{x \to 0} \cos \left(a x\right) \times {L}_{x \to 0} \cos \left(b x\right)$

or $\frac{1}{b} \times 1 \times 1 \times 1 = \frac{1}{b}$