OP Malhotra Class 12 Maths Solutions Chapter 9 Indeterminate Forms of Limits Ex 9(b)

Accessing OP Malhotra Maths Class 12 Solutions Chapter 9 Indeterminate Forms of Limits Ex 9(b) can be a valuable tool for students seeking extra practice.

Find by applying L’Hospital’s Rule the following limits :

Question 1.
\(\lim _{x \rightarrow 0} \frac{x e^x-\log (1+x)}{x^2}\)
Solution:

Question 2.
\(\lim _{x \rightarrow 0} \frac{x-\sin x \cos x}{x^3}\)
Solution:

Question 3.
\(\lim _{x \rightarrow 0} \frac{e^x \sin x-x-x^2}{x^3}\)
Solution:

Question 4.
\(\lim _{x \rightarrow \pi / 4} \frac{\sin \left(x+\frac{\pi}{4}\right)-1}{\log \sin 2 x}\)
Solution:

Question 5.
\(\lim _{x \rightarrow 0} \frac{e^x+e^{-x}+2 \cos x-4}{x^4}\)
Solution:

Question 6.
\(\lim _{x \rightarrow 0} \frac{e^x-e^{-x}-2 \log (1+x)}{x \sin x}\)
Solution:

Question 7.
\(\lim _{x \rightarrow \frac{1}{2}} \frac{\cos ^2 \pi x}{e^{2 x}-2 e x}\)
Solution:

Question 8.
\(\lim _{x \rightarrow 0} \frac{\log \left(1-x^2\right)}{\log \cos x}\)
Solution:

Question 9.
\(\lim _{x \rightarrow 0} \frac{\tan x-x}{x^2 \tan x}\)
Solution:

Question 10.
\(\lim _{x \rightarrow 0} \frac{\sin x-\log \left(e^x \cos x\right)}{x \sin x}\)
Solution:

Question 11.
\(\lim _{x \rightarrow 0} \frac{\log \sin x}{\cot x}\)
Solution:

Question 12.
\(\lim _{x \rightarrow 0} \frac{\log \tan x}{\log x}\)
Solution:

Question 13.
\(\lim _{x \rightarrow 0} \frac{\log \tan 2 x}{\log \tan x}\)
Solution:

Question 14.
\(\lim _{x \rightarrow 0} \log (1-x) \cot \frac{\pi x}{2}\)
Solution:

Question 15.
\(\lim _{x \rightarrow 0}\left(\frac{1}{x^2}-\cot ^2 x\right)\)
Solution:

Question 16.
\(\lim _{x \rightarrow \frac{\pi}{2}}(\sin x)^{\tan x}\)
Solution:

Question 17.
\(\lim _{x \rightarrow 0}(\cot x)^{\sin 2 x}\)
Solution:

Question 18.
\(\lim _{x \rightarrow 0}(1+\sin x)^{\cot x}\)
Solution:

Question 19.
\(\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{1 / x^2}\)
Solution:

Question 20.
\(\lim _{x \rightarrow 0}\left(\frac{e^x-e^{-x}-2 x}{x-\sin x}\right)\)
Solution: