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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Limits

NDA 2022 · Mathematics · Limits

What is _ x arrow 1 x^3-1 x -1 equal to?

What is lim⁡x→1x3−1x−1\lim _{x \rightarrow 1} \frac{x^3-1}{\sqrt{x}-1} equal to?

Options

  1. A0
  2. B3
  3. C6
  4. DLimit does not exist

Correct answer

C. 6

Step-by-step solution

lim⁡x→1x3−1x−1=lim⁡x→1(x3−1)(x+1)x−1=lim⁡x→1(x−1)(x2+x+1)(x+1)x−1=(1+1+1)(1+1)=6\begin{aligned} & \lim _{x \rightarrow 1} \frac{x^3-1}{\sqrt{x}-1}=\lim _{x \rightarrow 1} \frac{\left(x^3-1\right)(\sqrt{x}+1)}{x-1} \\ & =\lim _{x \rightarrow 1} \frac{(x-1)\left(x^2+x+1\right)(\sqrt{x}+1)}{x-1} \\ & =(1+1+1)(1+1)=6\end{aligned}
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