Given the rational function Q(x)= 4x-1/x^2+7x+12. What is the domain of Q(x)? What is the vertical/horizontal Asymptotes

Answer 1

the denominator factors as (x+4)(x+3)

therefore the vertical asymptotes are at x=-4 and x=-3, and these values are NOT

in the domain...

So the domain is (-inifinity, -4) U (-4,-3) U (-3,infinity)

for long term behavior, divides everything by x^2, the highest power of x:

4/x - 1/x divided by (1 + 7/x + 12/x^2)

as x goes off to infinity, the function dissipates to zero.

so there is a vertical asymptote at y=0, which the function will cross at x=1/4