If length of perpendicular drawn from points of a curve to a straight line approaches zero along an infinite brar ch of the curve, the line is said to be an asymptote to the curve. For example, \( y \)-axis is an
\( \mathrm{P} \) asymptote to \( y=\ln x \& x \)-axis is an asymptote to \( y=e^{-x} \).
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Asymptotes parallel to \( x \)-axis :
If \( \lim _{x \rightarrow \infty} f(x)=e \) (a finite number) then \( y=e \) is an asymptote to \( y=f(x) \). Similarly if \( \lim _{x \rightarrow-\infty} f(x)=\alpha \), then
\( y=\alpha \) is also an asymptote.
Asymptotes parallel to \( y \)-axis :
If \( \lim _{x \rightarrow a} f(x)=\infty \) or \( \lim _{x \rightarrow a} f(x)=-\infty \), then \( x=a \) is an asymptote to \( y=f(x) \).
Area bounded by \( y=\frac{2 x}{x^{2}+1} \), it's asymptote and ordinates at points of extremum is equal to (in square
unit)
(A) \( \ln 2 \)
(B) \( 2 \ln 2 \)
(C) in 3
(D) \( 2 \ln 3 \)
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