Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Using the method of L' Hospital, evaluate the limit. <img alt="\lim _{x \rightarrow 4} \frac{15-5 \sqrt{5+x}}{1-\sqrt{5-x}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_

Using the method of L' Hospital, evaluate the limit. \lim _{x \rightarrow 4} \frac{15-5 \sqrt{5+x}}{1-\sqrt{5-x}}

Option: 1

\frac{5}{3}


Option: 2

\frac{10}{3}


Option: 3

-\frac{5}{3}


Option: 4

4


Answers (1)

best_answer

Apply the L' Hospital's Rule, when \lim _{x \rightarrow 0} \frac{f(x)}{g(x)}=\frac{0}{0}, \frac{\infty}{\infty}, etc. [an intermediate form] in the following way

- Differentiate \lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}, provided that the limit \lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)} exists.

- Otherwise, go on differentiating till the limit with the determinate form is achieved.

The provided limit is

\lim _{x \rightarrow 4} \frac{15-5 \sqrt{5+x}}{1-\sqrt{5-x}}=\frac{0}{0}

Here use the L' Hospital rule and also refer to the following formula.

\frac{d}{d x} x^{n}=n x^{n-1}

Now, derive the following: 

\begin{aligned} & \lim _{x \rightarrow 4} \frac{15-5 \sqrt{5+x}}{1-\sqrt{5-x}} \\ & =\lim _{x \rightarrow 4} \frac{\frac{d}{d x}(15-5 \sqrt{5+x})}{\frac{d}{d x}(1-\sqrt{5-x})} \\ & =\lim _{x \rightarrow 4}\left(\frac{-\frac{5}{2 \sqrt{5+x}}}{-\frac{(-1)}{2 \sqrt{5-x}}}\right) \quad\left[\frac{d}{d x} x^{n}=n x^{n-1}\right] \\ \end{aligned}

\begin{aligned} & =\frac{-\frac{5}{\sqrt{5+4}}}{\frac{1}{\sqrt{5-4}}} \\ & =-\frac{5}{3} \end{aligned}

Posted by

rishi.raj

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

Faridabad: 17-year-old girl preparing for JEE Main 2026 dies by suicide
anam-khan

NTA JEE Mains 2026 application closes at 9 am today; correction facility from December 1
anam-khan

JEE Main 2026 Registration LIVE: NTA session 1 correction facility opens on December 1; exam pattern, cut-offs

Latest Question