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 0} \frac{16^{x}-1}{e^{4 x}-1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5C

Using the method of L' Hospital, evaluate the limit. \lim _{x \rightarrow 0} \frac{16^{x}-1}{e^{4 x}-1}

Option: 1

-\frac{1}{\log _{2} e}


Option: 2

-\log _{2} e


Option: 3

\log _{e} 2


Option: 4

\log _{2} e


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 0} \frac{16^{x}-1}{e^{4 x}-1}=\frac{0}{0}

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

\begin{aligned} & \frac{d}{d x}\left(b^{x}\right)=b^{x} \ln b \\ & \frac{d}{d x}\left(e^{a x}\right)=a e^{a x} \end{aligned}

Now, derive the following:

$$ \begin{aligned} & \lim _{x \rightarrow 0} \frac{16^{x}-1}{e^{4 x}-1} \\ & =\lim _{x \rightarrow 0} \frac{\frac{d}{d x}\left(16^{x}-1\right)}{\frac{d}{d x}\left(e^{4 x}-1\right)} \\ & =\lim _{x \rightarrow 0} \frac{16^{x} \ln 16}{4 e^{4 x}} \\ & =\frac{16^{0} \times \ln 2^{4}}{4 \times e^{0}} \\ \end{aligned}

\begin{aligned} & =\frac{4 \ln 2}{4} \\ & =\ln 2 \\ & =\log 2 \end{aligned}

Posted by

Divya Prakash Singh

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

JEE Main 2026 application correction begins on jeemain.nta.nic.in; edit details by December 2
anam-khan

JEE Mains 2026 LIVE: NTA to conduct session 1 from January 21 to 30; exam pattern, syllabus
anam-khan

JEE Main 2026 correction window opens tomorrow; NTA allows exam city change

Latest Question