Question

Create a function which has the following properties:

a. It has a horizontal asymptote at y=2

b. It has a discontinuity at x=2 which is not a vertical asymptote

c. It has no other discontinuities or asymptotes

Explain how your answer satisfies the previous properties.

Answer 1

(a) 2 + 1/x
(b) (x-2)/(x-2)
(c) 1/x won't work, so 2+1/(1+x^2)

y = (2+1/(1+x^2))(x-2)/(x-2)
= (2x^3-4x^2+3x-6)/(x^3-2x^2+x-2)

see the graph at

http://www.wolframalpha.com/input/?i=%282x^3-4x^2%2B3x-6%29%2F%28x^3-2x^2%2Bx-2%29

Note that the graph is exactly the same as for

y = 2+1/(1+x^2)

except that it is undefined at x=2.