91Ó°ÊÓ

How do the roots of the equation \(a x^{2}+b x+c=0\) change when \(b\) and \(c\) retain constant values \((b \neq 0)\) and \(a\) tends to zero. \\{ns. One root tends to \(-\frac{c}{b}\) and the other root tends to \(\left.\pm \infty\right\\}\)

$$ \lim _{x \rightarrow+\infty} \frac{x^{b}}{a^{x}} $$

$$ \lim _{x \rightarrow \infty} \frac{(a x+1)^{n}}{x^{n}+A} \text { . Consider separately the cases when } n \text { is (1) positive integer, (2) negative integer, (3) zero } $$

$$ \text { Find the constants } \left.a \text { and } b \text { such that } \lim _{x \rightarrow \infty} \frac{x^{2}+1}{x+1}-a x-b=0 \quad \text { Ans. } a=1, b=-1\right\\} $$

$$ \text { If } \lim _{x \rightarrow \infty} \frac{x^{2}+1}{x+1}-a x-b=\infty, \text { find the value of } a \text { and } b \text { . } $$

$$ \text { If } \lim _{x \rightarrow \infty} \frac{x^{2}-1}{x+1}-a x-b=2, \text { find the values of } a \text { and } b \text { . } $$

$$ \text { If } \lim _{x \rightarrow \infty} \frac{x^{3}+1}{x^{2}+1}-a x-b=2 \text { , then find } a \text { and } b \text { . } $$

$$ \text { Find the constants } \left.a \text { and } b \text { such that } \lim _{x \rightarrow-\infty} \sqrt{x^{2}-x+1}-a x-b=0 \text { \\{ Ans. } a=-1, b=\frac{1}{2}\right\\} $$

$$ \text { If } \left.\lim _{x \rightarrow 0} \frac{a e^{x}+b e^{-x}+c x}{x^{3}}=1 \text { , find the values of } a, b, c \text { . \\{Ans. } a=3, b=-3, c=-6\right\\} $$

$$ \text { If } \lim _{x \rightarrow 0} \frac{\sin x+a e^{x}+b e^{-x}+c \ln (1+x)}{x^{3}} \text { is finite, find } a, b, c \text { and the value of the limit. } $$