How to find a if the function below doesn't have a vertical asymptotes. f(x)=(x2−ax+2)/(x−2)

Question

Asymptotes of function
How to find a if the function below doesn't have a vertical asymptotes.

f (x) = \frac{x^{2} - a x + 2}{x - 2}

Answer 1

A function like that has vertical asymptote when it's denominator is 0, that happens when x=2

f (x) = \frac{x^{2} - a x + 2}{x - 2}

But since we don't want f(x) to have a vertical asymptotes, then x−2 must be a factor of

x^{2} + a x + 2

x^{2} - a x + 2 = (x - 2) (x - b)

6 - 2 a = 0

f (x) = \frac{x^{2} - 3 x + 2}{x - 2}

f (x) = \frac{(x - 2) \cdot (x - 1)}{x - 2}

f (x) = (x - 1)

Answer 2

Hint: This can only happen if x−2 is a factor of the numerator, so it can "cancel" the x−2 in the denominator. What does

a

have to be for this to be true?

Answers (2)