79487
If \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\), then :
1 \(\mathrm{a}=1\) and \(\mathrm{b}=1\)
2 \(\mathrm{a}=1\) and \(\mathrm{b}=-1\)
3 \(a=1\) and \(b=-2\)
4 \(\mathrm{a}=1\) and \(\mathrm{b}=2\)
Explanation:
(C) : Given, \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1-a x^{3}-b x^{2}-a x-b}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}(1-a)-b x^{2}-a x-b+1}{x^{2}+1}\right]=2\) For the infinite limit exist, the coefficient of \(x^{3}\) must be zero. \(\therefore \quad 1-a=0\) \(a=1\) \(\lim _{x \rightarrow \infty}\left[\frac{-b x^{2}-x-b+1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{-b\left(x^{2}+1\right)-x+1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{-b\left(x^{2}+1\right)}{x^{2}+1}-\frac{x}{x^{2}+1}+\frac{1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[-b-\frac{x}{x^{2}+1}+\frac{1}{x^{2}+1}\right]=2\) \(-b-0+0=2\) \(b=-2\)
Karnataka CET-2000
Limits, Continuity and Differentiability
79488
If \(f(x)=\left|\begin{array}{ccc} \sin x & \cos x & \tan x \\ x^3 & x^2 & x \\ 2 x & 1 & 1 \end{array}\right|\), then \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}\) is :
79487
If \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\), then :
1 \(\mathrm{a}=1\) and \(\mathrm{b}=1\)
2 \(\mathrm{a}=1\) and \(\mathrm{b}=-1\)
3 \(a=1\) and \(b=-2\)
4 \(\mathrm{a}=1\) and \(\mathrm{b}=2\)
Explanation:
(C) : Given, \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1-a x^{3}-b x^{2}-a x-b}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}(1-a)-b x^{2}-a x-b+1}{x^{2}+1}\right]=2\) For the infinite limit exist, the coefficient of \(x^{3}\) must be zero. \(\therefore \quad 1-a=0\) \(a=1\) \(\lim _{x \rightarrow \infty}\left[\frac{-b x^{2}-x-b+1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{-b\left(x^{2}+1\right)-x+1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{-b\left(x^{2}+1\right)}{x^{2}+1}-\frac{x}{x^{2}+1}+\frac{1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[-b-\frac{x}{x^{2}+1}+\frac{1}{x^{2}+1}\right]=2\) \(-b-0+0=2\) \(b=-2\)
Karnataka CET-2000
Limits, Continuity and Differentiability
79488
If \(f(x)=\left|\begin{array}{ccc} \sin x & \cos x & \tan x \\ x^3 & x^2 & x \\ 2 x & 1 & 1 \end{array}\right|\), then \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}\) is :
79487
If \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\), then :
1 \(\mathrm{a}=1\) and \(\mathrm{b}=1\)
2 \(\mathrm{a}=1\) and \(\mathrm{b}=-1\)
3 \(a=1\) and \(b=-2\)
4 \(\mathrm{a}=1\) and \(\mathrm{b}=2\)
Explanation:
(C) : Given, \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1-a x^{3}-b x^{2}-a x-b}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}(1-a)-b x^{2}-a x-b+1}{x^{2}+1}\right]=2\) For the infinite limit exist, the coefficient of \(x^{3}\) must be zero. \(\therefore \quad 1-a=0\) \(a=1\) \(\lim _{x \rightarrow \infty}\left[\frac{-b x^{2}-x-b+1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{-b\left(x^{2}+1\right)-x+1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{-b\left(x^{2}+1\right)}{x^{2}+1}-\frac{x}{x^{2}+1}+\frac{1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[-b-\frac{x}{x^{2}+1}+\frac{1}{x^{2}+1}\right]=2\) \(-b-0+0=2\) \(b=-2\)
Karnataka CET-2000
Limits, Continuity and Differentiability
79488
If \(f(x)=\left|\begin{array}{ccc} \sin x & \cos x & \tan x \\ x^3 & x^2 & x \\ 2 x & 1 & 1 \end{array}\right|\), then \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}\) is :
79487
If \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\), then :
1 \(\mathrm{a}=1\) and \(\mathrm{b}=1\)
2 \(\mathrm{a}=1\) and \(\mathrm{b}=-1\)
3 \(a=1\) and \(b=-2\)
4 \(\mathrm{a}=1\) and \(\mathrm{b}=2\)
Explanation:
(C) : Given, \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1}{x^{2}+1}-(a x+b)\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}+1-a x^{3}-b x^{2}-a x-b}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{x^{3}(1-a)-b x^{2}-a x-b+1}{x^{2}+1}\right]=2\) For the infinite limit exist, the coefficient of \(x^{3}\) must be zero. \(\therefore \quad 1-a=0\) \(a=1\) \(\lim _{x \rightarrow \infty}\left[\frac{-b x^{2}-x-b+1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{-b\left(x^{2}+1\right)-x+1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[\frac{-b\left(x^{2}+1\right)}{x^{2}+1}-\frac{x}{x^{2}+1}+\frac{1}{x^{2}+1}\right]=2\) \(\lim _{x \rightarrow \infty}\left[-b-\frac{x}{x^{2}+1}+\frac{1}{x^{2}+1}\right]=2\) \(-b-0+0=2\) \(b=-2\)
Karnataka CET-2000
Limits, Continuity and Differentiability
79488
If \(f(x)=\left|\begin{array}{ccc} \sin x & \cos x & \tan x \\ x^3 & x^2 & x \\ 2 x & 1 & 1 \end{array}\right|\), then \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}\) is :