14.2. Graphs to Find Limits http://www.ck12.org
Functions like the one above with discontinuities, asymptotes and holes require you to have a very solid understand-
ing of how to evaluate and interpret limits.
Atx=a, the function is undefined because there is a vertical asymptote. You would write:
f(a) =DNE,limx→af(x) =DNE
Atx=b, the function is defined because the filled in circle represents that it is the height of the function. This
appears to be at about 1. However, since the two sides do not agree, the limit does not exist here either.
f(b) = 1 ,limx→bf(x) =DNE
Atx=0, the function has a discontinuity in the form of a hole. It is as if the point( 0 ,− 2. 4 )has been lifted up
and placed at( 0 , 1 ). You can evaluate both the function and the limit at this point, however these quantities will not
match. When you evaluate the function you have to give the actual height of the function, which is 1 in this case.
When you evaluate the limit, you have to give what the height of the function is supposed to be based solely on the
neighborhood around 0. Since the function appears to reach a height of -2.4 from both the left and the right, the limit
does exist.
f( 0 ) = 1 ,limx→ 0 f(x) =− 2. 4
Atx=c, the limit does not exist because the left and right hand neighborhoods do not agree on a height. On the
other hand, the filled in circle represents that the function is defined atx=cto be -3.
f(c) =− 3 ,limx→cf(x) =DNE
Atx→∞you may only discuss the limit of the function since it is not appropriate to evaluate a function at infinity
(you cannot findf(∞)). Since the function appears to increase without bound, the limit does not exist.
xlim→∞f(x) =DNE
Atx→−∞the graph appears to flatten as it moves to the left. There is a horizontal asymptote aty=0 that this
function approaches asx→−∞.
xlim→−∞f(x) =^0
When evaluating limits graphically, your main goal is to determine whether the limit exists. The limit only exists
when the left and right sides of the functions meet at a specific height. Whatever the function is doing at that point
does not matter for the sake of limits. The function could be defined at that point, could be undefined at that point,
or the point could be defined at some other height. Regardless of what is happening at that point, when you evaluate
limits graphically, you only look at the neighborhood to the left and right of the function at the point.
Example A