Question

How do you find the asymptotes for `f(x)=(1-5x) /( 1+2x)`?

Answer 1

vertical asymptote `x = -1/2`
horizontal asymptote `y = -5/2`

Explanation:

vertical asymptotes occur when the denominator of a rational function tends to zero.

To find the equation

solve 1 + 2x = 0 `rArr x = -1/2`

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

If the degree of the numerator an denominator are equal then the equation can be found by taking the ratio of leading coefficients

in this case they are both of degree 1

I'll rewrite f(x) to assist in finding leading coefficients

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

equation of asymptote: `y = -5/2`

here is the graph of f(x)
graph{(1-5x)/(1+2x) [-10, 10, -5, 5]}