# How do I find the limit as x approaches infinity of a trigonometric function?

## ${\lim}_{x \to \infty} \frac{{x}^{2} \csc 3 x \tan 6 x}{\cos 7 x {\cot}^{2} x}$

Oct 4, 2016

The limit does not exist...

#### Explanation:

First consider:

$f \left(x\right) = \frac{\csc 3 x \tan 6 x}{\cos 7 x {\cot}^{2} x}$

The various constituent trigonometric functions have periods:

$\frac{2 \pi}{3} , \frac{\pi}{6} , \frac{2 \pi}{7} , \pi$

The least common multiple of these is $2 \pi$. Hence $f \left(x\right)$ has period $2 \pi$ or a factor thereof. In fact we can tell that it has period exactly $2 \pi$ since $7$ is prime.

When $x = \frac{\pi}{14}$ we find that $\cos 7 x = \cos \left(\frac{\pi}{2}\right) = 0$ and all of the other trigonometric functions are non-zero.

If we take a small interval around $x = \frac{\pi}{14}$ then $\cos 7 x$ changes sign while the other trigonometric functions retain their signs. Hence $f \left(x\right)$ changes sign in a small interval around $x = \frac{\pi}{14}$.

Since $\cos 7 x$ is in the denominator, this means that $f \left(x\right)$ has a vertical asymptote with different signs on either side of the asymptote at $x = \frac{\pi}{14} + 2 n \pi$ for any integer $n$.

Next note that ${x}^{2} \to \infty$ as $x \to \infty$ (or $x \to - \infty$).

Note also that all the trigonometric functions are continuous on their various domains.

Hence:

$\frac{{x}^{2} \csc 3 x \tan 6 x}{\cos 7 x {\cot}^{2} x}$

is unbounded and takes every value in $\left(- \infty , \infty\right)$ repeatedly as $x \to \infty$ (or $x \to - \infty$). It definitely does not converge to a limit.