KC Sinha Mathematics Solution Class 11 Chapter 28 Limit at Infinity and Infinite Limits and Existence of Limits Exercise 28.1

 Exercise 28.1

Page no 28.8

Question 1

Find (i) $\lim _{x \rightarrow \infty} x^{3}$

(ii) $\lim _{x \rightarrow \infty} 2^{x}$

(iii) $\lim _{x \rightarrow \infty} 3^{-x}$

(iv) $\lim _{x \rightarrow \infty} \frac{1}{(x-1)^{2}}$

Question 2

Find (i) $\lim _{x \rightarrow \infty} \frac{x-1}{x+1}$

(ii) $\lim _{x \rightarrow \infty} \frac{2 x-1}{x+2}$

(iii) $\lim _{x \rightarrow \infty} \frac{x-1}{x}$

Question 3

Find (i) $\lim _{x \rightarrow \infty} \frac{2 x^{3}+2 x^{2}+1}{3 x^{3}+x+2}$

(ii) Find $\lim _{x \rightarrow \infty} \frac{x^{2}+x+5}{2 x^{3}+x^{2}}$

(iii) $\lim _{x \rightarrow \infty} \frac{x^{4}+7 x^{3}+46 x+a}{x^{4}+6}, a \neq 0$

Question 4

Find $\lim _{x \rightarrow \infty} \frac{x^{2}+a x+b}{x^{2}+p x+q}$, where $a, b, p, q$ are non-zero real numbers

Question 5

Find $\lim _{x \rightarrow \infty} \frac{(x+1)^{10}+(x+2)^{10}+(x+3)^{10}}{x^{10}+(x+3)^{10}}$

Question 6

Find $\lim _{x \rightarrow \infty} \frac{(3 x-1)(4 x-2)}{(x+8)(x-1)}$

Question 7

Let $f(x)=\frac{a x+b}{x+1}, \lim _{x \rightarrow 0} f(x)=2$ and $\lim _{x \rightarrow \infty} f(x)=1$. Prove that $f(-2)=1$.

Question 8

Find $\lim _{x \rightarrow \infty} \frac{a x^{p}+b x^{p-2}+c}{d x^{q}+e x^{q-3}+k}$

Where $a>0, d>0, p>0, q>0$ and $a, b, c, d, e$ and $k$ are constants.

Question 9

Find $\lim _{x \rightarrow \infty} \frac{a_{0} x^{n}+a_{1} x^{n-1}+\ldots+a_{n}}{b_{0} x^{m}+b_{1} x^{m-1}+\ldots+b_{m}}, a_{0}>0, b_{0}>0$

Question 10

Find $\lim _{x \rightarrow \infty}\left(\frac{1}{x^{2}}+\frac{2}{x^{2}}+\ldots+\frac{x}{x^{2}}\right)$

Question 11

Find $\lim _{n \rightarrow \infty}\left(\frac{1^{3}}{n^{4}}+\frac{2^{3}}{n^{4}}+\frac{3^{3}}{n^{4}}+\ldots+\frac{n^{3}}{n^{4}}\right)$

Question 12

(i) Find $\lim _{n \rightarrow \infty}\left(\frac{1^{2}+1}{n^{3}}+\frac{2^{2}+2}{n^{3}}+\frac{3^{2}+3}{n^{3}}+\ldots+\frac{n^{2}+n}{n^{3}}\right)$

(ii) Find $\lim _{n \rightarrow \infty} \frac{1.2+2 \cdot 3+\ldots+n \cdot(n+1)}{n^{3}}$

Question 13

Find  $\lim _{n \rightarrow \infty}\left(\frac{1+2+3+\ldots+n}{n+2}-\frac{n}{2}\right)$

Question 14

Find $\lim _{n \rightarrow \infty} \frac{1-2+3-4+\ldots+(2 n-1)-2 n}{\sqrt{n^{2}+1}}$

Question 15

Find $\lim _{n \rightarrow \infty} \frac{(n+2) !+(n+1) !}{(n+3) !}$

Question 16

Find $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+x}-x\right)$

Question 17

Find $\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+a x}-\sqrt{x^{2}-a x}\right)$

Question 18

Find $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+x+1}-x\right)$

Question 19

 Find $\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+x-1}-x\right)$

Question 20

Find $\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}-8 x}+x\right)$

Question 21

Find $\lim _{x \rightarrow-\infty}\left(\sqrt{4 x^{2}+7 x}+2 x\right)$

Question 22

Find $\lim _{n \rightarrow \infty} \frac{\left(\sqrt{n^{2}+1}+n\right)^{2}}{\left(n^{6}+1\right)^{1 / 3}}$

Question 23

Find $\lim _{n \rightarrow \infty}\left(1+\frac{1}{2}+\frac{1}{4}+\ldots+\frac{1}{2^{n}}\right)$

Question 24

Find $\lim _{n \rightarrow \infty} \frac{2^{n}-1}{2^{n}+1}$