Question

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

Answer 1

`x=6` and `y = 1`

Explanation:

3 types of asymptotes : Vertical, horizontal and oblique.

Vertical :

The straight line `x=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.
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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 :

Note : `y=1` is in green.