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  4. displaystyle lim x arrow ∞((x2+1/x+1)-a x-b),(a, b ∈ R)=0 text . Then

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Q. $\displaystyle\lim _{x \rightarrow \infty}\left(\frac{x^2+1}{x+1}-a x-b\right),(a, b \in R)=0 \text {. Then }$

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Solution:

$ \displaystyle\lim _{x \rightarrow \infty} \frac{x^2+1-a x(x+1)-b(x+1)}{x+1}=0 $
$= \displaystyle\lim _{x \rightarrow \infty} \frac{(1-a) x^2-(a+b) x+1-b}{x+1}=0$
for the limit to exist,
$ 1-a=0 \text { and }-(a+b)=0$
$\therefore a=1, b=-1$