Rational Functions?

Question

Find the vertical, horizontal and oblique asymptotes for the rational function below, if any. Explain what you are doing step by step.

R(x) = (x + 5)/(x^2 + 1)

Answer 1

R(x) = (x+5)/ (x^2 +1)
the curve looks like an inverted bell
max point (x=0.099 , y = 5.0495)
intersection point (-5, 0) . .and . .(0,5)
it is not exactly asymptote . . . almost tangent with y = 0.021

Answer 2

lim R(x) = 0 using L'Hospital/s Rule
x-->infinity
So the x-axis is a horizontal asymptote
The denominator is always positive so there are no vertical asymptotes. The function is continuous over the reals. There are no oblique asymptotes.

Answer 3

Let's find the domaine of definition of the function R(x).
Find the domain of each function given
f(x) = x + 5
h(x) = x^2 + 1

Notice that h(x) is always positive in the space R.

the domaine of f(x) is R.
So, the domaine of R(x) is R.

As x --> oo, R(x) takes values close to zero, and the graph approaches the line horizontal line y = 0. This line is called the horizontal asymptote.