Expert: Paul Klarreich - 10/1/2006

Question
I am studying College Algebra, and this question has got me stumped.  the question is this:
Describe what happens to the graph of a function as the value of the independent variable approaches its vertical asymptote.

I'm not really sure how to word this, can you help me out?  

Answer
Oliver Asks in Category Calculus ...
 
Subject:  The Graph of a Rational Function
Private:  no
 
Question:  I am studying College Algebra, and this question has got me stumped.  the question is this:
Describe what happens to the graph of a function as the value of the independent variable approaches its vertical asymptote.

I'm not really sure how to word this, can you help me out?  
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Hi, Oliver,

Actually, the answer is contained in the question, as it so often is.  If the graph of a function has a vertical asymptote at  x = a, then typically:

A. the function is undefined at x = a, i.e. it does not touch the asymptote.
B. as x approaches a from the left, the graph becomes higher and higher OR it becomes lower and lower.  We would say that  

lim   f(x) = infinity, (gets higher) OR
x->a-

lim   f(x) = - infinity  (gets lower)
x->a-

So you draw the curve zooming up or down, but not touching the vertical line x = a.

C. as x approaches a from the RIGHT, the graph becomes higher and higher OR it becomes lower and lower.  We would say that  

lim   f(x) = infinity, (gets higher) OR
x->a+

lim   f(x) = - infinity  (gets lower)
x->a+

So you draw the curve zooming up or down, but not touching the vertical line x = a, and this part is on the right side.

But be careful to distinguish 'algebraic' language from 'geometric' language.  

So you would talk about the function using algebraic phrasing like:

lim   f(x) = infinity
x->a-

and you would talk about its graph using geometric phrasing, like:

The graph gets higher and higher without any bound, but does not touch the vertical asymptote.

I hope this helps.