Chapter 1. Section 6
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C. Bellomo, revised 16-Aug-08
Section 1.6 – Limits at Infinity
Recall:
• You have already studied rational functions, that is ()
() ()
p
x
fx qx
=.
• You should already be able to identify the domain of this function, in other words, the values of x
that are/are not allowed for inputs of f.
• Q: How do we find any undefined points in the domain?
A: Set the denominator equal to zero and solve for x.
• These points, if they are not removable discontinuities, are called infinite discontinuities, and the
limit of the function there would tend to plus or minus infinity.
• These types of infinite discontinuities are also called vertical asymptotes.
• Example. For 1
() 2
fx x
=− What is the limit as x tends to 2 from the right? From the left?
First note that the domain is restricted when
20 2xx−=⇒=.
Also note that the top does not go to 0 at x = 2.
x = 2 is an infinite discontinuity.
2
lim ( )
xfx
+
→
=∞
2
lim ( )
xfx
−
→
=−∞
Horizontal Asymptotes:
• Think of an asymptote in general being an invisible line that a function ‘tends’ to if you were to keep
drawing. You never quite get there.
• For example, in the above function, the graph (as x tends to 2) gets closer and closer to the vertical
line at x = 2 but never quite reaches it. And it will not cross over, either.
• The above example function has another asymptote that is horizontal… If you were to let x get larger
and larger (
x
→±∞) the function would get closer and closer to an ‘invisible’ horizontal line.
• Q: What value does the function tend to as
x
→±∞? What is the equation of this line?
A: It tends to 0, or the equation y = 0.
