# How do you use the Squeeze Theorem to find lim Tan(4x)/x as x approaches infinity?

Sep 27, 2015

There is no limit of that function as $x \rightarrow \infty$

#### Explanation:

I know of no version of the squeeze theorem that can be use to show that this limit does not exist.

Observe that as $4 x$ approaches and odd multiple of $\frac{\pi}{2}$, $\tan \left(4 x\right)$ becomes infinite (in the positive or negative direction depending on the direction of approach).

So every time $x \rightarrow \text{odd} \times \frac{\pi}{8}$ the numerator of $\tan \frac{4 x}{x}$ becomes infinite while the denominator approaches a (finite) limit. Therefore there is no limit of $\tan \frac{4 x}{x}$ as $x \rightarrow \infty$

Although the Squeeze theorem is not helpful, it may be possible to use a boundedness theorem to prove this result.
That is, it may be possible to show that for large $x$, we have $\left\mid \tan \frac{4 x}{x} \right\mid \ge f \left(x\right)$ for some $f \left(x\right)$ that has vertical asymptotes where $\tan \frac{4 x}{x}$ has them.

For reference, here is the graph of $f \left(x\right) = \tan \frac{4 x}{x}$

graph{tan(4x)/x [-3.91, 18.59, -4.87, 6.37]}