What is oblique asymptote of graph y = (x^2-6x+m)/(2x-4) if m=9?

Friday, November 16, 2018

What is oblique asymptote of graph y = (x^2-6x+m)/(2x-4) if m=9?

The line
$\displaystyle{y}={a}{x}+{b}$ is the oblique asymptote for the function $\displaystyle{y}=\frac{{{{x}}^{{2}}-{6}{x}+{m}}}{{{2}{x}-{4}}}$ if there exists
$\displaystyle{a}=\lim_{{{x}\to\pm\infty}}\frac{{y}}{{x}}$ and $\displaystyle{b}=\lim_{{{x}\to\pm\infty}}{\left({y}-{a}\cdot{x}\right)}$ .

You need
first to evaluate a, at $\displaystyle{m}={9}$ , such that:

$\displaystyle{a}=\lim_{{{x}\to\pm\infty}}{\left({{x}}^{{2}}-{6}{x}\right.}$
+ 9)/(x(2x - 4))

You need to force factor $\displaystyle{{x}}^{{2}}$ to numerator and $\displaystyle{x}$ to
denominator, such that:

$\displaystyle{a}=\lim_{{{x}\to\pm\infty}}\frac{{{{x}}^{{2}}{\left({1}-\frac{{6}}{{x}}+\frac{{9}}{{{x}}^{{2}}}\right)}}}{{{{x}}^{{2}}{\left({2}-\right.}}}$
4/x))