GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 5

Question

$\lim _{x \rightarrow 2} \frac{\sqrt{x}-\sqrt{2}}{x-2}$

• A. $0$
• B. $\sqrt{2}$
• C. $\frac{1}{2 \sqrt{2}}$
• D. $\frac{1}{\sqrt{2}}$

   

Answer 1

Given : $\lim_{x -> 2} (\sqrt{x} - \sqrt{2})/(x-2)$

 

If we try to put value x = 2 in the given limits. We can see this is $0/0$ form.

If we have 0/0 form we can simply apply L-Hospital Rule.

 

NOTE : IN L-HOSPITAL’S RULE, DIFFERENTIATE THE NUMERATOR AND DENOMINATOR, UNTIL 

0/0 FORM GETS ELIMINATED.

 

NOTE : $\frac{\mathrm{d} }{\mathrm{d} x} (\sqrt{x}) = 1 / (2\sqrt{x})$

 

Apply L- Hospital Rule : 

$(1/2\sqrt{x}$ –  0)/(1 – $ 0)$ 

$(1/2\sqrt{x})/(1)$ 

 

Now, apply the limits. We will get : $1/(2\sqrt{2})$

 

Option C, is Correct Answer.