Question

How do you find vertical, horizontal and oblique asymptotes for `y = (x + 1)/(x - 1)`?

Answer 1

vertical asymptote x = 1
horizontal asymptote y = 1

Explanation:

Vertical asymptotes occur when the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.

solve : x - 1 = 0 → x = 1 is the asymptote

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

divide all terms on numerator and denominator by x

`(x/x + 1/x )/(x/x - 1/x ) = (1 + 1/x)/(1 - 1/x )`

As x →∞ , `1/x → 0`

`rArr y = 1/1 = 1 " is the asymptote "`

Oblique asymptotes occur when the degree of the numerator is greater than the degree of the denominator. This is not the case here , hence there are no oblique asymptotes.

Here is the graph of the function.
graph{(x+1)/(x-1) [-10, 10, -5, 5]}