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Limits of Trigonometric Functions - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Trigonometric Limits is considered one the most difficult concept.

  • 613 Questions around this concept.

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QuestionEASY

\lim_{x\rightarrow \frac{\pi }{2}}\: \: \frac{\cot x-\cos x}{\left ( \pi -2x \right )^{3}}equals:

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\lim_{x\rightarrow 0}\frac{\sin \left ( \pi \cos ^{2}x \right )}{x^{2}}   is equal to :

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if f(x) is continuous and f(\frac{9}{2})=\frac{2}{9}, then \lim_{x\rightarrow 0}f\left ( \frac{1-\cos 3x}{x^{2}} \right )  is equal to :

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\lim_{x\rightarrow 2}\left ( \frac{\sqrt{1-\cos \left \{ 2(x-2) \right \}}}{x-2} \right )

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Concepts Covered - 1

Trigonometric Limits

Trigonometric Limits

In the trigonometric limit, apart from using the method of direct substitution, factorization and rationalization (same as given in algebraic limits), we can use the following formulae.

\mathbf{\text{(i)}\;\;\;\lim_{x\rightarrow 0}\frac{\sin x}{x}=1}

\\\mathbf{\text{(ii)}\;\;\;\lim_{x\rightarrow 0}\frac{\tan x}{x}=1} \\\\\text{\;\;\;\;\;\;\;\;As}\;\;\lim_{x\rightarrow 0}\frac{\tan x}{x}=\lim_{x\rightarrow 0}\frac{\sin x}{x}\times\frac{1}{\cos x}\\\\\text{\;\;\;\;\;\;\;\;}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\lim_{x\rightarrow 0}\frac{\sin x}{x}\times\lim_{x\rightarrow 0}\frac{1}{\cos x}=1\times 1
\mathbf{\text{(iii)}\;\;\;\lim_{x\rightarrow a}\frac{\sin (x-a)}{x-a}=1}\\\\\text{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;As}\;\;\lim_{x\rightarrow a}\frac{\sin (x-a)}{x-a}=\lim_{h\rightarrow 0}\frac{\sin ((a+h)-a)}{(a+h)-a}\\\\\text{\;\;\;\;\;\;\;\;}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\lim_{h\rightarrow 0}\frac{\sin h}{h}\\\\\text{\;\;\;\;\;\;\;\;}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=1

\mathbf{\text{(iv)}\;\;\;\lim_{x\rightarrow a}\frac{\tan (x-a)}{x-a}=1}

\\(v)\;\;\lim_{x \rightarrow a} \frac{\sin (f(x))}{f(x)}=1,\;\;if\;\lim_{x\rightarrow a}f(x)=0\\ \,\,\,\,\,\,\,\:\:Similarly, \;\;\;\lim_{x\rightarrow a}\frac{\tan(f(x))}{f(x)}=1,\;\;if\;\lim_{x\rightarrow a}f(x)=0

\mathbf{\text{(vi)}\;\;\;\lim_{x\rightarrow 0}\cos x=1}

\\\mathbf{\text{(vii)}\;\;\;\lim_{x\rightarrow 0}\frac{\sin^{-1} x}{x}=1}\\ \\\text{\;\;\;\;\;\;\;\;\;As}\;\;\lim_{x\rightarrow 0}\frac{\sin^{-1} x}{x}=\lim_{y\rightarrow 0}\frac{y}{\sin y}\;\;\;\;\;\;\;\;[\because \sin^{-1}x=y]\\\\\text{\;\;\;\;\;\;\;\;}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=1

\\\mathbf{\text{(viii)}\;\;\;\lim_{x\rightarrow 0}\frac{\tan^{-1} x}{x}=1}

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Trigonometric Limits

Study other Related Concepts

Limits of Trigonometric Functions Current Topic

Limit
Left-Hand Limits and Right-Hand Limits
Algebra of Limits
Limit of Indeterminate Form and Algebraic limit
Limit of Algebraic function
Limit Using Expansion
Sandwich Theorem
Limits of Trigonometric Functions
Exponential and Logarithmic Limits
Limits of the form (1 power infinity)

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Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

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