Answer

Verified

379.8k+ views

**Hint**: By substituting the value of $ x=1 $ in the limit expression we get $ \dfrac{0}{0} $ form which is an indeterminate form so we apply L'Hopital's rule to find the value of limit in which we differentiate numerator and denominator with respect to x and then substitute the value of $ x=1 $ in the differentiated expression. In this way, we will get the value of the limit.

**:**

__Complete step-by-step answer__In the above problem, we have given:

$ G\left( x \right)=\sqrt{25-{{x}^{2}}} $

The limit that we have to evaluate in the above problem is:

$ \underset{x\to 1}{\mathop{\lim }}\,\dfrac{G\left( x \right)-G\left( 1 \right)}{x-1} $

Substituting the value of $ G\left( x \right)=\sqrt{25-{{x}^{2}}} $ in the above limit expression we get,

$ \underset{x\to 1}{\mathop{\lim }}\,\dfrac{\sqrt{25-{{x}^{2}}}-G\left( 1 \right)}{x-1} $

Finding the value of G(1) by substituting the value of $ x=1 $ in $ G\left( x \right)=\sqrt{25-{{x}^{2}}} $ we get,

$ \begin{align}

& G\left( 1 \right)=\sqrt{25-{{\left( 1 \right)}^{2}}} \\

& \Rightarrow G\left( 1 \right)=\sqrt{24} \\

\end{align} $

Now, substituting $ G\left( 1 \right)=\sqrt{24} $ in the limit expression we get,

$ \underset{x\to 1}{\mathop{\lim }}\,\dfrac{\sqrt{25-{{x}^{2}}}-\sqrt{24}}{x-1} $

Applying limit on the expression written inside the limit by substituting $ x=1 $ we get,

$ \begin{align}

& \dfrac{\sqrt{25-{{\left( 1 \right)}^{2}}}-\sqrt{24}}{1-1} \\

& =\dfrac{\sqrt{24}-\sqrt{24}}{0} \\

& =\dfrac{0}{0} \\

\end{align} $

As you can see that the value of limit is in the form of $ \dfrac{0}{0} $ which is an indeterminate form so we are going to apply L’Hospital’s rule to find the value of limit. In L'Hospital's rule we differentiate numerator and denominator with respect to x and then substitute the value of $ x=1 $ .

Let us assume $ f\left( x \right)=\sqrt{25-{{x}^{2}}}-\sqrt{24} $ and $ g\left( x \right)=x-1 $ and then applying L’Hospital’s rule we get,

$ \underset{x\to 1}{\mathop{\lim }}\,\dfrac{f(x)}{g(x)}=\underset{x\to 1}{\mathop{\lim }}\,\dfrac{f'(x)}{g'(x)} $ …………. Eq. (1)

Now, solving f’(x) and g’(x) we get,

$ f'\left( x \right)=\dfrac{-\left( 2x \right)}{2\sqrt{25-{{x}^{2}}}} $

2 will be cancelled out from numerator and denominator we get,

$ f'\left( x \right)=\dfrac{-\left( x \right)}{\sqrt{25-{{x}^{2}}}} $

$ g'\left( x \right)=1 $

Substituting the value of $ f'\left( x \right)\And g'\left( x \right) $ in eq. (1) we get,

$ \begin{align}

& \underset{x\to 1}{\mathop{\lim }}\,\dfrac{f'(x)}{g'(x)} \\

& =\underset{x\to 1}{\mathop{\lim }}\,\dfrac{-x}{\sqrt{25-{{x}^{2}}}} \\

\end{align} $

Substituting $ x=1 $ in the above limit expression we get,

$ \begin{align}

& \dfrac{-\left( 1 \right)}{\sqrt{25-{{\left( 1 \right)}^{2}}}} \\

& =-\dfrac{1}{\sqrt{24}} \\

\end{align} $

We can write 24 which is in the square root as $ 6\times 4 $ we get,

$ \begin{align}

& -\dfrac{1}{\sqrt{4\left( 6 \right)}} \\

& =-\dfrac{1}{2\sqrt{\left( 6 \right)}} \\

\end{align} $

From the above calculations, we have evaluated the limit as $ -\dfrac{1}{2\sqrt{\left( 6 \right)}} $ .

**So, the correct answer is “Option A”.**

**Note**: The point to be noted here is that while applying L’Hospital’s rule in the above solution, we have differentiated numerator and denominator just once and then substitute the value of $ x=1 $ and got the value of limit but in some questions after differentiating the numerator and denominator and then we put the limit in the differentiated expression again we are getting $ \dfrac{0}{0} $ form so we have to again differentiate numerator and denominator of the first differentiation result and then substitute the limit. We have to do this differentiation till we have got the determinant value.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Describe the poetic devices used in the poem Aunt Jennifers class 12 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

And such too is the grandeur of the dooms We have imagined class 12 english CBSE

Where is the endless fountain of immortal drink pouring class 10 english CBSE

What I want should not be confused with total inactivity class 12 english CBSE

Why was the Moral Rearmament Army welcomed at the class 12 english CBSE

Difference Between Plant Cell and Animal Cell