Expert: Paul Klarreich - 11/8/2006

Question
Hi paul my question is,

Explain why the function f(x) = {1 if x is less than or equal to 0 and (1/x) if x is greater than 0, has a vertical asymptote but no vertical tangent.

Answer
Questioner:  Tom
Category:  Calculus
 
Subject:  Sketching a graph with asymptotes.
Question:  Hi paul my question is,

Explain why the function f(x) = {1 if x is less than or equal to 0 and (1/x) if x is greater than 0, has a vertical asymptote but no vertical tangent.
......................................................
Hi, Tom,  

       |  1,  if  x <= 0
f(x) =  |
       |  1/x, if x > 0

I think this is just a matter of terminology.  Generally, a line x=a is:

A. a VA if the values of f(x) become infinite as x->a.

B. a VT if the values of f'(x) become infinite as x->a, but f(x) is continous at x=a

For example, the semicircle   y = sqrt(4-x^2) has a VT at x = 2, but not a VA.

In your case, the values of f(x) on the right become infinite as x -> 0, and so do the values of f'(x).  So x=0 is an asymptote, but not a vertical tangent, because f(x) is not continuous at x = 0.

How does this sound?