Math Doubts

Evaluate $\displaystyle \Large \lim_{x \,\to\, 0} \normalsize \dfrac{1-\sqrt{1-\tan x}}{\sin x}$

$x$ is a literal number and represents an angle of the right angled triangle as well. It formed a function in ratio form with trigonometric functions sine and tangent. The value of the function is required to evaluate when the value of $x$ tends to $0$.

$\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{1-\sqrt{1-\tan x}}{\sin x}$

01

Substitute x is equal to 0

$= \,\,\,$ $\dfrac{1-\sqrt{1-\tan (0)}}{\sin (0)}$

$= \,\,\,$ $\dfrac{1-\sqrt{1-0}}{0}$

$= \,\,\,$ $\dfrac{1-\sqrt{1}}{0}$

$= \,\,\,$ $\dfrac{1-1}{0}$

$= \,\,\,$ $\dfrac{0}{0}$

The function become indeterminate when the value of $x$ approaches zero. So, the limit problem must be solved in another method.

02

Apply Rationalizing method

Use rationalising method by multiplying both numerator and denominator of the function by the rationalising factor of the numerator.

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{1-\sqrt{1-\tan x}}{\sin x}$ $\times$ $\dfrac{1+\sqrt{1-\tan x}}{1+\sqrt{1-\tan x}}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{ (1-\sqrt{1-\tan x}) \times (1+\sqrt{1-\tan x}) }{ \sin x (1+\sqrt{1-\tan x})}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{ {(1)}^2 -{(\sqrt{1-\tan x})}^2}{\sin x (1+\sqrt{1-\tan x})}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{1 -(1-\tan x)}{\sin x (1+\sqrt{1-\tan x})}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{1-1+\tan x}{\sin x (1+\sqrt{1-\tan x})}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\require{cancel} \dfrac{\cancel{1}-\cancel{1}+\tan x}{\sin x (1+\sqrt{1-\tan x})}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{\tan x}{\sin x (1+\sqrt{1-\tan x})}$

03

Use Reciprocal identity of sine and cosine

There is a $\sin x$ term in denominator and it can be cancelled by expressing $\tan x$ in terms of ratio of $\sin x$ to $\cos x$ by the reciprocal identity of sine and cosine functions.

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{\dfrac{\sin x}{\cos x}}{\sin x (1+\sqrt{1-\tan x})}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{\sin x}{\sin x \times \cos x \times (1+\sqrt{1-\tan x})}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\require{cancel} \dfrac{\cancel{\sin x}}{\cancel{\sin x} \times \cos x \times (1+\sqrt{1-\tan x})}$

$=\,\,\,$ $\displaystyle \Large \lim_{x \,\to\, 0}$ $\dfrac{1}{\cos x (1+\sqrt{1-\tan x})}$

04

Substitute x is equal to 0

Now substitute $x$ is equal to zero to evaluate the function when $x$ tends to zero.

$=\,\,\,$ $\dfrac{1}{\cos (0) (1+\sqrt{1-\tan (0)})}$

$=\,\,\,$ $\dfrac{1}{1 \times (1+\sqrt{1-0})}$

$=\,\,\,$ $\dfrac{1}{1 \times (1+\sqrt{1})}$

$=\,\,\,$ $\dfrac{1}{1 \times (1+1)}$

$=\,\,\,$ $\dfrac{1}{1 \times 2}$

$=\,\,\,$ $\dfrac{1}{2}$

It is the required solution for this limit problems of the calculus mathematics.

Latest Math Topics
Latest Math Problems
Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved