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

Practicing OP Malhotra Maths Class 12 Solutions Chapter 9 Indeterminate Forms of Limits Ex 9(a) is the ultimate need for students who intend to score good marks in examinations.

S Chand Class 12 ICSE Maths Solutions Chapter 9 Indeterminate Forms of Limits Ex 9(a)

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

Question 1.
\(\lim _{x \rightarrow 2} \frac{x^3-8}{x^2-4}\)
Solution:

Question 2.
\(\lim _{x \rightarrow 3} \frac{x^3-x^2-18}{x-3}\)
Solution:

Question 3.
\(\lim _{x \rightarrow 0} \frac{\tan ^{-1} x}{x}\)
Solution:

Question 4.
\(\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x}\)
Solution:

Question 5.
\(\lim _{x \rightarrow 1} \frac{x-1}{2 x^2-7 x+5}\)
Solution:

Question 6.
\(\lim _{x \rightarrow 0} \frac{(1+x)^5-1}{(1+x)^3-1}\)
Solution:

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

Question 8.
\(\lim _{x \rightarrow a} \frac{\sin x-\sin \alpha}{x-\alpha}\)
Solution:

Question 9.
\(\lim _{x \rightarrow 0} \frac{a^x-b^x}{e^x-1}\)
Solution:

Question 10.
\(\lim _{\alpha \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos \alpha}{\alpha-\frac{\pi}{4}}\)
Solution:

Question 11.
\(\lim _{x \rightarrow a} \frac{(x+2)^{5 / 3}-(a+2)^{5 / 2}}{x-a}\)
Solution:

Question 12.
\(\lim _{x \rightarrow 0} \frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{x}\)
Solution:

Question 13.
If f(1) = 1. f'(1) = 2, then find \(\lim _{x \rightarrow 1} \frac{\sqrt{f(x)}-1}{\sqrt{x}-1}\)
Solution:

Question 14.
\(\lim _{x \rightarrow 0} \frac{2^x-1}{\sqrt{1+x}-1}\)
Solution:

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

Question 16.
\(\lim _{x \rightarrow 0} \frac{(1+x)^n-1}{x}\)
Solution:

Question 17.
\(\lim _{x \rightarrow 0} \frac{1-x^{-1 / 3}}{1-x^{-2 / 3}}\)
Solution:

Question 18.
\(\lim _{x \rightarrow 0} \frac{1-\cos x}{x}\)
Solution:

Question 19.
\(\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}\)
Solution:

Question 20.
\(\lim _{x \rightarrow 0} \frac{\tan 2 x-x}{3 x-\sin x}\)
Solution:

Question 21.
\(\lim _{x \rightarrow 0} \frac{e^{x^2}-\cos x}{x^2}\)
Solution:

Question 22.
\(\lim _{x \rightarrow 0} \frac{x \cos x-\log (1+x)}{x^2}\)
Solution:

Question 23.
\(\lim _{x \rightarrow 0} \frac{\cos a x-\cos b x}{x^2}\)
Solution:

Question 24.
\(\lim _{x \rightarrow 0} \frac{x \cos x-\sin x}{x^2 \sin x}\)
Solution:

Question 25.
\(\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{\log (1+x)}{x^2}\right)\)
Solution:

Question 26.
\(\lim _{x \rightarrow 0} \frac{y-\tan ^{-1} y}{y-\sin y}\)
Solution:

Question 27.
\(\lim _{x \rightarrow 0^{+}} \log _{\sin 2 x} \sin x\)
Solution:

Question 28.
\(\lim _{x \rightarrow 0}\left(\frac{1}{x^2}-\frac{\cot x}{x}\right)\)
Solution: