Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2014
Exam
NDA
Year
2014
Subject
Mathematics
Chapter
Limits

NDA 2014 · Mathematics · Limits

What is _ x arrow 0 x 1- x equal to?

What is lim⁡x→0x1−cos⁡x\lim _{x \rightarrow 0} \frac{x}{\sqrt{1-\cos x}} equal to?

Options

  1. A2\sqrt{2}
  2. B−2-\sqrt{2}
  3. C12\frac{1}{\sqrt{2}}
  4. DLimit does not exist

Correct answer

A. 2\sqrt{2}

Step-by-step solution

lim⁡x→0x1−cos⁡x×1+cos⁡x1+cos⁡x=lim⁡x→0x(1+cos⁡x)1−cos⁡2x=lim⁡x→0x(1+cos⁡x)sin⁡x=lim⁡x→0xsin⁡x×lim⁡x→01+1=1×2=2.\begin{aligned} & \lim _{x \rightarrow 0} \frac{x}{\sqrt{1-\cos x}} \times \frac{\sqrt{1+\cos x}}{\sqrt{1+\cos x}} \\ & =\lim _{x \rightarrow 0} \frac{x(\sqrt{1+\cos x})}{\sqrt{1-\cos ^2 x}}=\lim _{x \rightarrow 0} \frac{x \sqrt{(1+\cos x)}}{\sin x} \\ & =\lim _{x \rightarrow 0} \frac{x}{\sin x} \times \lim _{x \rightarrow 0} \sqrt{1+1}=1 \times \sqrt{2}=\sqrt{2}.\end{aligned}
Practice Limits on Quantrex Academy →

More from Limits

What is _ x→1 x^ (n^2-1) -1 x^ (n+1) -1 equal to, where n > 1 is a natural number ? NDA 2026What is _ x→0 10^ x -1 x equal to ? NDA 2026Consider the following for the two (02) items that follow: Let the function f(x) = x^2 +… NDA 2025For the following three (03) items: Consider the function f(x) = x|x| . What is _ x → -1… NDA 2025For the following two (02) items: Let f(x) = cases (1 - 2x)/(x^2), & x 0 cases What is _… NDA 2025For the following two (02) items: Let f(x) = cases (1 - 2x)/(x^2), & x 0 cases What is _… NDA 2025Consider the following for the two (02) items that follow: Let the function f(x) = [x]… NDA 2025Consider the following for the two (02) items that follow: Let the function f(x) = [x]… NDA 2025Consider the following for the two (02) items that follow: Let the function f(x) = x^2… NDA 2025What is ( θ- θ) equal to? x arrow (π)/(2) NDA 2024Which one of the following is correct regarding _ x arrow 3 (|x-3|)/(x-3) ? NDA 2024Let f(x)=|x|+1 and g(x)=[x]-1 , where [.] is the greatest integer function. Let… NDA 2024Let f(x) a^ x-1 b^ x-1 and g(x)=x-1 . What is _ x arrow 1 (f(x)-1)/(g(x)) equal to? NDA 2023Let f(x)=|x| and g(x)=[x]-1 , where [.] is the greatest integer function. Let… NDA 2023Let f(x) a^ x-1 b^ x-1 and g(x)=x-1 . What is _ x arrow 1 f(x)^ (1)/(g(x)) equal to? NDA 2023What is _ x arrow 1 (x^9-1)/(x^3-1) equal to? NDA 2023 Full Limits list Limits JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs