Correct Answer - Option 3 : 1
Concept:
If f(x)/g(x) is in the from 0/0 or ∞/∞ Where x = a plugs in , then :
\(lim_{x→a}\) f(x)/g(x) = \(lim_{x→a}\) f'(x)/g'(X)
Calculation:
Given:
f(4) = 4, f '(4) = 1, then \(\displaystyle\lim_{x \rightarrow 4} \dfrac{2- \sqrt{f(x)}}{2-\sqrt x}\) from(0/0)
Now by L-Hospital Rule ,
⇒ \(lim_{x→4}\) \(\frac{0-\frac{1}{2\sqrt{f(n)}}×f'(n)}{0-\frac{1}{2\sqrt{n}}}\)
⇒ \(lim_{x→4}\) \(\frac{f'(x)}{\sqrt{f(x)}}×\sqrt{x}\)
⇒ \(\frac{f'(4)}{\sqrt{f(4)}}×\sqrt{4}\)
⇒ \(\frac{1×2}{\sqrt{4}}\)
⇒ 1
