If F(x)={9-x²}, then what is {x→1}{lim}F(x)-F(1)/x-1 equal to?

Correct Answer is : −122-{1}{2{2}}−221

Correct Answer is : −122-{1}{2{2}}−221

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UPSC NDA I 2017 Mathematics Solved Paper

Question : 65 of 120

Marks: +1, -0

F(x)=\sqrt{9-x^{2}}

\underset{x→1}{lim}\frac{F(x)-F(1)}{x-1}

equal to?

Solution:

Concept:

The limit expression is precisely the definition of the derivative of

F(x)

x=1

Explanation:

We have

\displaystyle \lim_{x\to 1}\frac{F(x)-F(1)}{x-1}=F'(1)

, provided the derivative exists.

Given

F(x)=\sqrt{9-x^{2}}

Differentiate using the chain rule:

F'(x)=\frac{1}{2\sqrt{9-x^{2}}}\cdot (-2x)=\frac{-x}{\sqrt{9-x^{2}}}

Substitute

x=1

F'(1)=\frac{-1}{\sqrt{9-1^{2}}}=\frac{-1}{\sqrt{8}}=-\frac{1}{2\sqrt{2}}

Answer:

-\frac{1}{2\sqrt{2}}

(Option C).

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