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  • Help me answer: - Limit , continuity and differentiability - JEE Main-2

If f(1)=1,f'(1)=2\; then\;      is

  • Option 1)

    2

  • Option 2)

    4

  • Option 3)

    1

  • Option 4)

    1/2

 

Answers (1)

As we learnt in 

L - Hospital Rule -

In \:the\:form\:of\:\:\;\frac{0}{0}\:\:and\:\:\frac{\infty }{\infty }\:\:\:we\:differentiate\:\:\frac{N^{r}}{D^{r}}\:\:separately.


\Rightarrow \lim_{x\rightarrow a}\:\:\frac{f(x)}{g(x)}=\lim_{x\rightarrow a}\:\:\frac{f'(x)}{g'(x)}

- wherein

\lim_{x\rightarrow a}\:\:\frac{\frac{d}{dx}\:f(x)}{\frac{d}{dx}\:g(x)}


Where \:\:f(x)\:\:and\:\:g(x)=0

 

 If  f\left ( 1 \right )=1

f'\left ( 1 \right )=2

Then  \lim_{x\rightarrow 1} \frac{\sqrt{f\left ( x \right )-1}}{\sqrt{x}-1}

\Rightarrow \lim_{x\rightarrow 1} \frac{\frac{f' \left ( x \right )}{2\sqrt{f\left ( x \right )}}}{\frac{1}{2\sqrt{x}}}=\frac{f'\left ( x \right )}{\sqrt{f' \left ( x \right )}}\times \sqrt{x}

= \frac{f'\left ( 1 \right )}{\sqrt{\left ( 1 \right )}}\times 1=\frac{2}{1}=2

 


Option 1)

2

Correct

Option 2)

4

Incorrect

Option 3)

1

Incorrect

Option 4)

1/2

Incorrect

Posted by

Vakul

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