# What is the limit of  (x / cos (-3x)) as x approaches infinity?

Jul 8, 2016

the limit wiil alternate between $- \infty$ and $\infty$

#### Explanation:

${\lim}_{x \to \infty} \left(\frac{x}{\cos} \left(- 3 x\right)\right)$

$= {\lim}_{x \to \infty} \left(\frac{x}{\cos} \left(3 x\right)\right)$ as $\cos \psi = \cos \left(- \psi\right)$

$= \frac{1}{\cos \left(3 x\right)} {\lim}_{x \to \infty} x$

because cos (3 x) is bounded and periodic, ie $\cos \left(- 3 x\right) \in \left[- 1 , 1\right]$ it can be taken out of the limit

but note that, as $x \to \infty$, the limit wiil alternate between $- \infty$ and $\infty$ as the cosine term changes sign with frequency $\frac{3}{\pi}$