# Limit of tanx/x as x approaches 0

The limit of tanx/x as x approaches 0 is equal to 1, that is,  limx0 tanx/x = 1. Here we learn to find the limit of x/tanx when x tends 0.

The formula of the limit of tanx/x when x→0 is given by limx0 tanx/x = 1.

## Proof of limit tanx/x = 1 when x→0

First, we prove the limit of tanx/x is 1 when x0 using the limit formula of sinx/x as x0. As tanx = sinx/cosx, we can write

limx→0 $\dfrac{\tan x}{x}$

= limx→0 $\dfrac{\sin x}{x\cos x}$

= limx→0 $\dfrac{\sin x}{x}$ × limx→0 $\dfrac{1}{\cos x}$

= 1 × $\dfrac{1}{\cos 0}$ by the limit formula limx0 sinx/x = 1.

= 1 × $\dfrac{1}{1}$ as cos0=1.

= 1

So the limit of tanx/x as x approaches to 0 is equal to 1.

## Proof of limx→0 tanx/x = 1 by L’Hôpital’s rule

Let us now prove that the limit of tanx/x is equal to 1 when x tends to 0 by the L’Hôpital’s rule of limits. Note that

limx→0 tanx/x = tan0/0 = 0/0, so it is an intermediate form. Thus, using the L’Hôpital’s rule we obtain that

limx→0 $\dfrac{\tan x}{x}$

= limx→0 $\dfrac{\frac{d}{dx}(\tan x)}{\frac{d}{dx}(x)}$

= limx→0 $\dfrac{\sec^2 x}{1}$

= sec2 0

= 1, as the value of sec0 is 1.

So the value of tanx/x limit is equal to 1 when x tends to 0.

Solved Problems on Exponential Limits

Sum Rule of Limits

Product Rule of Limits

Quotient Rule of Limits

## FAQs

### Q1: What is the limit of tanx/x when x tends to 0?

Answer: The limit of tanx/x when x tends to 0 is equal to 1.