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NDA 2014
Exam
NDA
Year
2014
Subject
Mathematics
Chapter
Limits

NDA 2014 · Mathematics · Limits

If f(9)=9 and f^ (9)=4 , then what is _ x arrow 9 f(x) -3 x -3...

If f(9)=9f(9)=9 and f′(9)=4f^\prime(9)=4, then what is lim⁡x→9f(x)−3x−3\lim _{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3} equal to?

Options

  1. A36
  2. B9
  3. C4
  4. DNone of the above

Correct answer

C. 4

Step-by-step solution

4=lim⁡x→9f(x)−9x−9⇒4=lim⁡x→9[f(x)−3x−3lim⁡x→9[f(x)+3x+3]⇒lim⁡x→9f(x)−3x−3⇒lim⁡x→91f(x)f′(x)1x=4313=4\begin{aligned} & 4=\lim _{x \rightarrow 9} \frac{f(x)-9}{x-9} \\ & \Rightarrow 4=\lim _{x \rightarrow 9}\left[\frac{\sqrt{f(x)}-3}{\sqrt{x}-3} \lim _{x \rightarrow 9}\left[\frac{\sqrt{f(x)}+3}{\sqrt{x}+3}\right]\right. \\ & \Rightarrow \lim _{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3} \\ & \Rightarrow \lim _{x \rightarrow 9} \frac{\frac{1}{\sqrt{f(x)}} f^{\prime}(x)}{\frac{1}{\sqrt{x}}}=\frac{\frac{4}{3}}{\frac{1}{3}}=4\end{aligned}
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