Question

If $f\left( 9 \right)=9,f'\left( 9 \right)=0$, then \[\displaystyle \lim_{x \to 9}\dfrac{\sqrt{f\left( x \right)}-3}{\sqrt{x}-3}\] is equal to
A. 0
B. $f\left( 0 \right)$
C. $f'\left( 3 \right)$
D. $f\left( 9 \right)$
E. 1

Answer 1

Hint: To solve this question first we will find the value of $f'\left( x \right)$, then we will substitute the values $f\left( 9 \right)=9,f'\left( 9 \right)=0$ in the obtained equation. Then by simplifying the obtained equation we will get the desired answer.

Complete step by step answer:
We have been given that $f\left( 9 \right)=9,f'\left( 9 \right)=0$.
We have to find the value of $\displaystyle \lim_{x \to 9}\dfrac{\sqrt{f\left( x \right)-3}}{\sqrt{x}-3}$.
Now, let us first find the derivative of the function. Then we will get
\[\Rightarrow \displaystyle \lim_{x \to 9}\left( \dfrac{\sqrt{f\left( x \right)}-3}{\sqrt{x}-3} \right)\]
Now, we know that $\dfrac{d}{dx}\sqrt{x}=\dfrac{1}{2\sqrt{x}}$
So by applying the above formula to the obtained equation we will get
\[\Rightarrow \displaystyle \lim_{x \to 9}\left( \dfrac{\dfrac{f'\left( x \right)}{2\sqrt{f\left( x \right)}}-0}{\dfrac{1}{2\sqrt{x}}-0} \right)\]
Now, we need to apply the limit in the obtained equation. For this we will replace the x by 9 in the above obtained equation as given the limit $\displaystyle \lim_{x \to 9}$. Now, applying the $\displaystyle \lim_{x \to 9}$ to the above obtained equation we will get
\[\Rightarrow \left( \dfrac{\dfrac{f'\left( 9 \right)}{2\sqrt{f\left( 9 \right)}}}{\dfrac{1}{2\sqrt{9}}} \right)\]
Now, substituting the values given in the question $f\left( 9 \right)=9,f'\left( 9 \right)=0$ in the above obtained equation we will get
\[\Rightarrow \left( \dfrac{\dfrac{0}{2\sqrt{9}}}{\dfrac{1}{2\sqrt{9}}} \right)\]
Now, cancelling the common terms we will get
$\Rightarrow 0$
Hence we get that \[\displaystyle \lim_{x \to 9}\dfrac{\sqrt{f\left( x \right)}-3}{\sqrt{x}-3}\] is equal to 0.

So, the correct answer is “Option A”.

Note: Alternatively we can solve the question by using the concept of rationalization. We will rationalize the denominator to simplify the expression and then we directly substitute the value of limit given in the question. To evaluate algebraic limits we will use the factorization method.