Limit of tanx/x as x approaches infinity

The limit of tanx/x as x approaches infinity does not exist, that is, lim_x→∞ 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 lim_x→∞ tanx/x = undefined. Below theorem will be useful to prove this formula.

Theorem: If lim_x→c f(x) exists, then for every sequence {x_n} converging to c the limit lim_n→∞ f(x_n) 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 lim_x→∞ f(x) does not exist, we consider two sequences {x_n} and {y_n}, both converging to ∞, but f(x_n) and f(y_n) converge to different limits. Let us assume that

x_n = nπ

y_n = π/2 + nπ – 1/n²

Here, both x_n, y_n →∞, but we have that

lim_n→∞ f(x_n) = lim_n→∞ $\frac{\tan n\pi}{n \pi}$ = 0.

On the other hand,

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

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

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

ALSO READ:

limx→0 (ex-1)/x = 1	limx→∞ sinx/x = 0
limx→0 x/cosx = 0	limx→0 x/sinx = 1
limx→0 tanx/x = 1	Sum rule of limits
Product rule of limits	Quotient rule of limits
ε – δ definition of limit

FAQs