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

Question

Aurora Alvarado · Accepted Answer

There is no limit of that function as #xrarroo#

There isn't a squeeze theorem variation that I'm aware of that can be used to demonstrate the nonexistence of this limit.

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 [-18.59, -4.87, 6.37, -3.91]}

Ezra Adams · Answer

undefined To use the Squeeze Theorem to find the limit of Tan(4x)/x as x approaches infinity, we can compare it to two other functions with known limits.

First, we know that the limit of Tan(4x) as x approaches infinity is undefined. However, we can find the limits of the functions sin(4x)/x and -sin(4x)/x as x approaches infinity, which are both equal to zero.

Since -1 ≤ sin(4x)/x ≤ 1 for all x, we can use the Squeeze Theorem to conclude that the limit of Tan(4x)/x as x approaches infinity is also zero.