Question

Sketch the asymptotes, and graph the function. y = 5/x-2+3

Answer 1

Y=5/(x-2)+3

Vertical Asymptote:
x-2=0
Solving for x:
x-2+2=0+2
x=2

Horizontal Asymptote:
Lim x→-Infinite y = Lim x→-Infinite 5/(x-2)+3=5/(-Infinite-2)+3=5/(-Infinite)+3
Lim x→-Infinite y = 0+3→Lim x→-Infinite y = 3

Lim x→Infinite y = Lim x→Infinite 5/(x-2)+3=5/(Infinite-2)+3=5/(Infinite)+3
Lim x→Infinite y = 0+3→Lim x→Infinite y = 3

Horizontal Asymptote: y=3

Please, see the graph in the attached file.
Thanks.

Answer 2

The figure attached shows the graph of the function with the two asympotes.
The formulae of those asympotes are:
y = 3, and x = 2.

This is how you work to find them.

1) The function will have vertical asympotes where the limit grows indefinetly (approach +/- ∞).

2) That happens for x = 2, where the function is not defined but it grows indefinetly toward infinity (if you come from the right side) or toward negative infinity (if you come from the left side). So, the vertical asymptote is the line x = 2.

3) The function will have a horizontal asymptote if the function tends to a constant value as x approaches infinity or negative infinity.

4) In this case, you find that the limit of the function when x appraches - ∞ or +∞ is 3.


That results in one horizontal asymptote which is y = 3.