Limit of tanx/x as x approaches 0

The limit of tanx/x as x approaches 0 is equal to 1, that is,  lim_x→0 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 lim_x→0 tanx/x = 1.

Proof of limit tanx/x = 1 when x→0

First, we prove the limit of tanx/x is 1 when x→0 using the limit formula of sinx/x as x→0. As tanx = sinx/cosx, we can write

lim_x→0 $\dfrac{\tan x}{x}$

= lim_x→0 $\dfrac{\sin x}{x\cos x}$

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

= 1 × $\dfrac{1}{\cos 0}$ by the limit formula lim_x→0 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.

ALSO READ:

limx→0 (ex-1)/x = 1	limx→∞ sinx/x = 0
limx→0 x/cosx = 0	limx→0 x/sinx = 1

Proof of lim_x→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

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

lim_x→0 $\dfrac{\tan x}{x}$

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

= lim_x→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.

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