This document discusses evaluating one-sided limits when the function value at the limit point is undefined, represented as f(a) = 0/0. There are four possible scenarios for the limits as the input approaches a from the left and right: (1) both limits are -∞, (2) the left limit is -∞ and the right is ∞, (3) the left is ∞ and the right is -∞, or (4) both are ∞. The overall limit only exists in cases (1) and (4). To determine if the limit is positive or negative infinity, evaluate the function very close to a on the given side.

1. Evaluating One-Sided Limits
Tyler Murphy
September 15, 2014
lim
x!a
f(x) = 1
1 Solving a limit when f(a) =
n
0
As with any limit, your

2. rst attempt should always be to plug the value for a into the
function and compute f(a). However, if you come up with an answer that is of the form
f(a)
n
0
then we have to look at some other possibilities. Whever you get f(a) =
n
0
then
you have a verticle asymptote at that point. Concerning the limit at this point, there are
four (4) possible scenarios.
(1) lim
x!a
f(x) = 1 : lim
x!a+
= 1
(2) lim
x!a
f(x) = 1 : lim
x!a+
= 1
(3) lim
x!a
f(x) = 1 : lim
x!a+
= 1
(4) lim
x!a
f(x) = 1 : lim
x!a+
= 1
Note that in both cases (1) and (4) that the overall limit (double-sided) exists, while
it does not exist in scenarios (2) and (3).
So in this situation all that is really left to do is determine if the limit is positive in

3. nity
or negative in

4. nity. The basic method for this is to pick a number, call it a0, that is really
close to a and on the side you are interested in and determine if the value is a large positive
number or a large negative number. (This only works if the function is a polynomial, not
trigonometric)
1