Calculus- Early Transcendentals, 2021a

76 Limits
One-sided limits
Consider the following piecewise defined function:
Observe from the graph that as x gets closer and closer to 1 from the left, then f (x) approaches +1.
Similarly, as x gets closer and closer 1 from the right, then f (x) approaches +2. We use the following
notation to indicate this:
lim f (x)=1 and lim f (x)=2.
x→1− x→1 +
The symbol x → 1 − means that we only consider values of x sufficiently close to 1 which are less than 1.
Similarly, the symbol x → 1 + means that we only consider values of x sufficiently close to 1 which are
greater than 1.
Definition 3.2: Left and Right-Hand Limit (Useable Definition)
In general, we will write
lim f (x)=L,
x→a −
if we can make the values of f (x) arbitrarily close to L by taking x to be sufficiently close to a and
x less than a.Thisiscalledtheleft-hand limit of f (x) as x approaches a. Similarly, we write
lim f (x)=L,
x→a +
if we can make the values of f (x) arbitrarily close to L by taking x to be sufficiently close to a and
x greater than a. This is called the right-hand limit of f (x) as x approaches a.
We note the following fact:
lim
x→a
f (x)=L if and only if lim f (x)=L and lim f (x)=L.
x→a− x→a +
Or more concisely:
lim f (x)= lim f (x)=L.
x→a− x→a +
A consequence of this fact is that if the one-sided limits are different, then the two-sided limit lim f (x) x→a
does not exist, often denoted as: (DNE).