Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Find the limit of <img alt="\mathrm{ \lim _{x \rightarrow 0} \frac{1-(\cos x)(\cos 2 x)^{1 / 2}(\cos 3 x)^{1 / 3}}{x^2} }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20%

Find the limit of \mathrm{ \lim _{x \rightarrow 0} \frac{1-(\cos x)(\cos 2 x)^{1 / 2}(\cos 3 x)^{1 / 3}}{x^2} }
 

Option: 1

3


Option: 2

5


Option: 3

2


Option: 4

1


Answers (1)

best_answer

The key here is to use Taylor / Maclaurin expansions as \mathrm{x \rightarrow 0}

                                     \mathrm{ \cos x=1-\frac{x^2}{2}+o\left(x^2\right) }
and

                                   \mathrm{ (1+x)^n=1+n x+o(x) }

where \mathrm{o(f(x))} represents a function \mathrm{g(x)} such that \mathrm{g(x) / f(x) \rightarrow 0.}

The starting step (based on expansion (1) above) thus needs to be written like

             \mathrm{ \lim _{x \rightarrow 0} \frac{1-\left(1-\frac{x^2}{2}+o\left(x^2\right)\right)\left(1-2 x^2+o\left(x^2\right)\right)^{1 / 2}\left(1-\frac{9 x^2}{2}+o\left(x^2\right)\right)^{1 / 3}}{x^2} }

and the next step (using (2)) becomes

            \mathrm{ \lim _{x \rightarrow 0} \frac{1-\left(1-\frac{x^2}{2}+o\left(x^2\right)\right)\left(1-x^2+o\left(x^2\right)\right)\left(1-\frac{3 x^2}{2}+o\left(x^2\right)\right)}{x^2} }

Finally we have via multiplication

                      \mathrm{ \lim _{x \rightarrow 0} \frac{1-\left(1-3 x^2+o\left(x^2\right)\right)}{x^2}=3 }

Posted by

vinayak

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 2025: 11 from Kota coaching centres among 24 students with 100 percentile
anam-khan

Low JEE Main rank? Explore alternative paths for engineering admission
anam-khan

Gender-gap in JEE Main 2025: Female participation remains around 31%, overall pool drops by 7.09%

Latest Question