# Limit of tanx/x as x approaches infinity

The limit of tanx/x as x approaches infinity does not exist, that is, limx tanx/x is undefined. Here we learn to find the limit of tanx/x when x tends to infinity.

The formula of the limit of tanx/x when x approaches ∞ is given by limx tanx/x = undefined. Below theorem will be useful to prove this formula.

Theorem: If limxc f(x) exists, then for every sequence {xn} converging to c the limit limn f(xn) is always unique.

## What is the Limit of tanx/x when x→∞

Answer: The value of the limit of tanx/x when x→∞ is undefined.

Proof:

Let f(x) = $\dfrac{\tan x}{x}$.

In order to show limx f(x) does not exist, we consider two sequences {xn} and {yn}, both converging to ∞, but f(xn) and f(yn) converge to different limits. Let us assume that

xn = nπ

yn = π/2 + nπ – 1/n2

Here, both xn, yn →∞, but we have that

limn f(xn) = limn $\frac{\tan n\pi}{n \pi}$ = 0.

On the other hand,

limn f(yn) = limn $\frac{\tan (\pi/2 + n\pi-1/n^2)}{\pi/2 + n\pi-1/n^2}$ = ∞.

As both sequences f(xn) and f(yn) converge to different limits, by the above theorem we conclude the following:

The limit of tanx/x when x→∞ does not exist.