Calculus Homework Help -- How do you use limits to determine the behavior of the function?

Question

I need help on using limits to determine the behavior of the function at any discontinuity. I also need to use limits to determine the end behavior and to sketch the graph. How do I find the discontinuities, intercepts, and asymptotes of these questions?

1. f(x) = (x^3+2x^2-35x) / (x^2-x-20)
2. f(x) = (-x-3) / (2x^2+x-15)

I first factored everything. However, like #1, I got stuck after finding out that -4 was an asymptote. How should I come about finding these requirements of the problems?

Thank you!

Answer 1

1. f(x) = (x³ + 2x² - 35x) / (x² - x - 20)

f(x) = (x³ + 2x² - 35x) / (x² - x - 20)
f(x) = x(x + 7)(x - 5) / [(x + 4)(x - 5)]
f(x) = x(x + 7) / (x + 4)

The domain is all real numbers except x = -4, 5.

There is a vertical asmptote at x = -4 and a hole in the function at x = 5. The term (x - 5) was in both the numerator and denominator. It cancels but still leaves a hold in the domain. The function is undefined at x = 5.

If we look at the remaining function we have:

f(x) = x(x + 7) / (x + 4)
f(x) = (x² + 7x) / (x + 4)

Dividing we get:
f(x) = (x + 3) - 12/(x + 4)

Take the limit of f(x) and get:
y = x +3

This is a slant asymptote of the function.

The zeroes of the function are
x = 0, -7

f(x) < 0 on the interval (-∞, -7)
f(x) = 0 for x = -7
f(x) > 0 on the interval (-7, -4)
f(x) is undefined for x = -4
f(x) < 0 on the interval (-4, 0)
f(x) = 0 for x = 0
f(x) > 0 on the interval (0, 5)
f(x) is undefined for x = 5
f(x) > 0 on the interval (5, ∞)

Answer 2

x^3 -a million = (x-a million)(x^2+x+a million) so expression turns into (x-a million) (x^2+x+a million)/(x-a million) = x^2+x+a million sub x=a million into x^2+x+a million = 3 x-25 = (sqrt[x]-5)(sqrt[x]+5) expression = a million/(sqrt[x]+5 ) sub x=25 to get a million / (5+5) = a million/10