R D Sharma Solutions for Chapter: Limits, Exercise 13: MULTIPLE CHOICE QUESTIONS (MCQs)

Evaluate $\underset{x\to \infty }{\lim }\frac{\sqrt{x^2-1}}{2x+1}$

Evaluate $\underset{n\to \infty }{\lim }\left\{\frac{1}{1·3}+\frac{1}{3·5}+\frac{1}{5·7}+\dots +\frac{1}{(2n+1)(2n+3)}\right\}$

Find $n$ if $\underset{x\to 1}{\lim }\frac{x+x^2+x^3+\dots +x^n-n}{x-1}=5050$.

Evaluate $\underset{n\to \infty }{\lim }\frac{1-2+3-4+5-6+\dots +(2n-1)-2n}{\sqrt{n^2+1}+\sqrt{n^2-1}}$

Evaluate $\underset{x\to \frac{\pi }{4}}{\lim }\frac{4\sqrt{2}-(\cos x+\sin x{)}^5}{1-\sin 2x}$

$\underset{x\to \infty }{\lim }\frac{\left|x\right|}{x}\text{ is equal to }$

$\underset{x\to 0}{\lim }\frac{\left|\sin x\right|}{x}\text{ is }$

If $f\left(x\right)=\left\{\begin{array}{c}\frac{\sin \left[x\right]}{\left[x\right]},\left[x\right]\ne 0\\ 0 ,\left[x\right]=0\end{array}\right.$ where $[.]$  denotes the greatest integer function, then find $\underset{x\to 0}{\lim }f\left(x\right)$.