Question

How do you find the asymptotes for `y = (7x-5)/(2-5x)`?

Answer 1

The asymptotes are `x=2/5` vertical asymptote
`y=-7/5` horizontal asymptote

Explanation:

Take the limit of y as x approaches `oo`

`lim_(x->oo) y=lim_(x->oo) (7x-5)/(-5x+2)=lim_(x->oo) (7-5/x)/(-5+2/x)=-7/5`

`x=-7/5`

Also if you solve for x in terms of y,

`y=(7x-5)/(-5x+2)`

`y(-5x+2)=7x-5`

`-5xy+2y=7x-5`

`2y+5=7x+5xy`

`2y+5=x(7+5y)`

`x=(2y+5)/(5y+7)`

take now the limit of x as y approaches `oo`

`lim_(y->oo) x=lim_(y->oo) (2y+5)/(5y+7)=lim_(y->oo) (2+5/y)/(5+7/y)=2/5`

`y=2/5`

kindly see the graph.

graph{y=(7x-5)/(-5x+2)[-20,20,-10,10]}

have a nice day!