Question

State the vertical asymptote of the rational function. f(x) =((x-9)(x+7))/(x^2-4)

Answer 1

There is a vertical asymptote for the rational function at x = −7. Set the denominator equal to 0 and solve for x.

x + 7 = 0 → x = −7

Explanation:

Answer 2

`D:x^2-4\\ot=0\\ D:x^2\\ot=4\\ D:x\\ot=-2 \wedge x\\ot =2\\\\ \displaystyle \lim_(x\to-2^-)((x-9)(x+7))/(x^2-4)=\\ ((-2-9)(-2+7))/((-2^-)^2-4)=(-11\cdot5)/(4^+-4)=(-55)/(0^+)=-55\cdot\infty=-\infty\\ ((-2-9)(-2+7))/((-2^+)^2-4)=(-11\cdot5)/(4^--4)=(-55)/(0^-)=-55\cdot(-\infty)=\infty`


`\displaystyle \lim_(x\to2^-)((x-9)(x+7))/(x^2-4)=\\ ((2-9)(2+7))/((2^-)^2-4)=(-7\cdot9)/(4^--4)=(-63)/(0^-)=-63\cdot(-\infty)=\infty\\ ((2-9)(2+7))/((2^+)^2-4)=(-7\cdot9)/(4^+-4)=(-63)/(0^+)=-63\cdot\infty=-\infty\\`

So, the vertical asymptotes are
`x=\pm 2`