Question

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

Answer 1

There is no limit of that function as `xrarroo`

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 `4x` approaches and odd multiple of `pi/2`, `tan(4x)` becomes infinite (in the positive or negative direction depending on the direction of approach).

So every time `x rarr "odd" xx pi/8` the numerator of `tan(4x)/x` becomes infinite while the denominator approaches a (finite) limit. Therefore there is no limit of `tan(4x)/x` as `xrarroo`

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 `abs(tan(4x)/x) >= f(x)` for some `f(x)` that has vertical asymptotes where `tan(4x)/x` has them.

For reference, here is the graph of `f(x) = tan(4x)/x`

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