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

1 Answer
Jim H
Sep 27, 2015

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]}

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