How do you find all the asymptotes for function y = x/(x-6) y=xx6?

1 Answer
Jul 6, 2015

x=6x=6 and y = 1y=1

Explanation:

3 types of asymptotes : Vertical, horizontal and oblique.

Vertical :

The straight line x=ax=a is a vertical asymptote for f(x) if lim_(x->a)f(x)=+-oo

Let f(x) = x/(x-6)

The domain of f(x) is ]-oo;6[uu]6;+oo[ <=> 6 is a forbidden value

We generally find a vertical asymptote in a rational function, when the value of denominator is equal to 0 (except if numerator is zero in the same time)

Look at f(x) when we are close to 6.

lim_(x->6^-)f(x) = -oo and lim_(x->6^+)f(x)=+oo

We comply with the starting conditions : x=6 is a vertical asymptote.


Horizontal :

The straight line y=b is a horizontal asymptote for f(x) if lim_(x->+-oo)f(x)=b, (b in RR)

To find easily the limit when x go to +-oo, we will modify f(x) :

AA x in RR-{6}, f(x) = x/(x-6) = (x-6+6)/(x-6) = 1*cancel((x-6)/(x-6)) + 6/(x-6) = 1 + 6/(x-6)

Limit calculation :

lim_(x->+oo)f(x) = lim_(x->+oo) 1 + 6/(x-6) = 1 + lim_(x->+oo) 6/(x-6)

And lim_(x->+oo) 6/(x-6) = 6xxlim_(x->+oo) 1/(x-6) = 6xxlim_(x->+oo) 1/x = 0

Then :
lim_(x->+oo)f(x) = 1 + 0 = 1

Do the same reasoning when x->-oo

Therefore : y=1 is a horizontal asymptote for f(x)

Oblique :

No oblique asymptote because lim_(x->+-oo)f(x) = 1

To sum up, look at the graph below :

enter image source here

Note : y=1 is in green.