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Question: How do you compute the limit of \(\dfrac{{\tan x}}{x}as\,x \to 0\)?...

How do you compute the limit of tan⁡xxas x→0\dfrac{{\tan x}}{x}as\,x \to 0?

Explanation

Solution

In order to determine the above limit ,rewrite the expression using tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}} and use the result lim⁡x→0sin⁡xx=1\dfrac{{\lim }}{{x \to 0}}\dfrac{{\sin x}}{x} = 1 to find the required answer.

Formula used:
cos⁡(0)=1\cos (0) = 1
lim⁡x→0sin⁡xx=1\dfrac{{\lim }}{{x \to 0}}\dfrac{{\sin x}}{x} = 1

Complete step by step solution:
We are given lim⁡x→0tan⁡xx\dfrac{{\lim }}{{x \to 0}}\dfrac{{\tan x}}{x},
Now as we know that tan⁡x\tan xcan be written as sin⁡xcos⁡x\dfrac{{\sin x}}{{\cos x}},so our expression
becomes
lim⁡x→0sin⁡xcos⁡xx\dfrac{{\lim }}{{x \to 0}}\dfrac{{\dfrac{{\sin x}}{{\cos x}}}}{x}
We can rearrange the expression as
lim⁡x→0sin⁡xx.1cos⁡x\dfrac{{\lim }}{{x \to 0}}\dfrac{{\sin x}}{x}.\dfrac{1}{{\cos x}}
Now one of the properties of limits is that limits of the products in other words something times something is the same as the product of the limits so We can take limits of each of those factors .
Here we are going to have

0}}\dfrac{1}{{\cos x}}} \right)$$ As we know that the limit of the $\dfrac{{\sin x}}{x}$ is equal to 1. $$ = (1) \times \left( {\dfrac{{\lim }}{{x \to 0}}\dfrac{1}{{\cos x}}} \right)$$ And to find the limit of $\dfrac{1}{{\cos x}}$simply replace x with 0. $$ = (1) \times \left( {\dfrac{1}{{\cos (0)}}} \right)$$ Since from the trigonometric table $$\cos (0) = 1$$

= (1) \times \left( {\dfrac{1}{1}} \right) \\
= 1 \\

$\therefore \dfrac{{\lim }}{{x \to 0}}\dfrac{{\tan x}}{x} = 1$ **Therefore, the limit of $\dfrac{{\tan x}}{x}as\,x \to 0$ is equal to 1.** **Note:** 1.Limit :You and your companions choose to meet at some spot outside. Is it essential that every one of your companions is living in a similar spot and stroll on a similar street to arrive at that place? Actually no, not generally. All companions come from various pieces of the city or nation to meet at that one single spot. It would appear that intermingling of various components to a solitary point. Mathematically, it resembles an intermingling of a function to a specific value. It is an illustration of cutoff points. Cut- off points show how a few functions are limited. The function watches out for some worth when its breaking point moves toward some value. 2\. Don’t forget to cross-check your answer.