a) Can the graph of y=f(x) intersect a vertical asymptote? Can it intersect a horizontal asymptote B)…..?

Question

A) Can the graph of y = f(x) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Please illustrate your answer by sketching relevant graphs. 
B) How many horizontal asymptotes can the graph of y = f(x) have? Sketch graphs to demonstrate the different possibilities.

Gus Gulgowski · Accepted Answer

A), no , it can't intersect the vertical asymptote, by definition of asymptote. The vertical asymptote indicates that it goes to infinity as it gets closer to a certain point x_0 . f(x) is not defined in x_0 therefore it can't touch the asymptote.
If it were a horizontal or oblique asymptote, it could intersect it before it tends to infinity.
B) It can have 2 horizontal asymptotes, one for x that goes to plus infinity and the other for x that goes to minus infinity. However you could have that the two asymptotes are the same, for example in the case of a symmetric function. (if the two limits for +infinity and - infinity are the same you get the same asymptotes)

Anonymous · Answer

if r(x) = -1/(x + 1)^2, then the function has a vertical asymptote at x = -1, but since the denominator is raised to an even power, it is convergent (both sides either go towards - infinity). There are no horizontal asymptotes. if r(x) = -1/(x - 1)^2, then the function has a vertical asymptote at x = 1, and it is also convergent. There are no horizontal asymptotes.

Anonymous · Answer

Yes, a graph may intersect a VA. Suppose f(a) = L and limit of f as x approaches a from the left is also L. But the limit of f as x approaches a from the right is positive or negative infinity. Then by definition, x = a is a VA, but x intersect the line x = a at the point (a, L).

Graciela Roob DDS

a) Can the graph of y=f(x) intersect a vertical asymptote? Can it intersect a horizontal asymptote B)…..?

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