Question

How do you find the vertical, horizontal or slant asymptotes for `f(x)= (x^2-5x+6)/ (x^2-8x+15)`?

Answer 1

`"Asymptote "->" "x=5`

`lim_(x->+-oo) f(x)= 1`

Explanation:

Given:`" "f(x)=(x^2-5x+6)/(x^2-8x+15)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Investigate potential simplification by factoring

`f(x)=((x-3)(x-2))/((x-3)(x-5)) = (x-2)/(x-5)`

At `x=5` the denominator becomes zero so the expression is undefined. Thus the asymptote is at `x=5`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
As `x` tends to `oo` then the constants are of no consequence.

`lim_(x->+-oo) f(x)= lim_(x->+-oo) (x-2)/(x-5) -> 1`