Step By Step Calculus » 13.4 - Slant Asymptotes

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Synopsis
If a function y=f(x)y=f(x) approximates to an equation of a line y=m^{\pm}x+b^{\pm}y=m^{\pm}x+b^{\pm} as x\to\pm\infty,x\to\pm\infty, then y=m^{\pm}x+b^{\pm}y=m^{\pm}x+b^{\pm} is called the slant asymptotes of y=f(x).y=f(x). More formally,
A line m^{+ }x+b^{+}m^{+ }x+b^{+} is a slant asymptote for ff as x\rightarrow +\infty x\rightarrow +\infty if
\displaystyle \lim_{x\rightarrow + \infty }\left[ f(x)-m^{+ }x-b^{+ }\right] =0.
\displaystyle \lim_{x\rightarrow + \infty }\left[ f(x)-m^{+ }x-b^{+ }\right] =0.
Similarly, a line m^{- }x+b^{-}m^{- }x+b^{-} is a slant asymptote for ff as x\rightarrow -\infty x\rightarrow -\infty if
\displaystyle \lim_{x\rightarrow - \infty }\left[ f(x)-m^{- }x-b^{- }\right] =0.
\displaystyle \lim_{x\rightarrow - \infty }\left[ f(x)-m^{- }x-b^{- }\right] =0.
The term oblique asymptote is a synonym for slant asymptote.
The formulae for calculating m^{\pm}m^{\pm} and b^{\pm}b^{\pm} are as follows:
\displaystyle m^{\pm }=\lim_{x\rightarrow \pm \infty }f^{\prime }(x)\displaystyle m^{\pm }=\lim_{x\rightarrow \pm \infty }f^{\prime }(x) and \displaystyle b^{\pm }=\lim_{x\rightarrow \pm \infty }\left( f(x)-m^{\pm }x\right) . \displaystyle b^{\pm }=\lim_{x\rightarrow \pm \infty }\left( f(x)-m^{\pm }x\right) .
If f(x)f(x) is a rational function, then slant asymptotes can be obtained by using long division to divide the numerator by the denominator.