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UPSC NDA Math Model Paper 5

Question : 113 of 120

Marks: +1, -0

Let

f(2)=2

f'(2)=2.

\lim\limits_{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}

is given by

Solution:

Given: f(2) = 2 and f’(2) = 2.

To find:

\lim\limits_{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}=?

Check the form by putting x = 2

\lim\limits_{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2}=\frac{2(2)-2(2)}{2-2}=\frac{0}{0}

Apply

L

' hospital's rule,

\lim\limits_{x \rightarrow 2} \frac{x f(2)-2 f(x)}{x-2} = \lim\limits_{x \rightarrow 2} \frac{\frac{d}{dx}(x f(2)-2 f(x))}{\frac{d}{dx}(x-2)}

= \lim\limits_{x \rightarrow 2} \frac{f(2)-2 f'(x)}{1}

Differentiate the numerator and differentiate the denominator, we get

Now, take limit at

x=2

\lim\limits_{x \rightarrow 2}=f(2)-2 f'(2)

=2-2(2)=-2

Option (3) is correct.

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