What are the asymptotes and removable discontinuities, if any, of #f(x)=((3x -6)/(x^2 -5x + 6))#?

1 Answer
Mar 25, 2016

removable discontinuity at x = 2
vertical asymptote x = 3
horizontal asymptote y = 0

Explanation:

First step is to factor the function.

#( 3(x - 2 ))/((x - 3 )(x - 2 )) =( 3cancel((x-2)))/((x-3)cancel((x-2))#

There is a removable discontinuity at x = 2 .

f(x) is now f(x) # = 3/(x-3) #

Vertical asymptotes occur as the denominator tends to zero. To find the equation let the denominator equal zero.

solve : x - 3 = 0 → x = 3 is the asymptote

Horizontal asymptotes occur as #lim_(x→±∞) f(x) → 0 #

divide all terms by x

# (3/x)/(x/x - 3/x) = (3/x)/(1- 3/x) #

As x → ∞ , # 3/x → 0 , y = 0" is the asymptote "#

Here is the graph of the function.
graph{(3x-6)/(x^2-5x+6) [-10, 10, -5, 5]}