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One-sided Limits
In order to calculate a limit at a point, we need to have an interval around that point. However, at some points, such as end points, this is not possible. For endpoints we can only find an interval on one side of the point. Instead, we can use the information that we are provided on that interval, in order to calculate a one-sided limit.
Definition: Right-hand Limit We say that L is the right-hand limit of f (x) at x 0 , written
lim x→x+ 0
f (x) = L
if for every number > 0, there exists a corresponding number δ > 0 such that for all x x 0 < x < x 0 + δ =⇒ |f (x) − L| < .
Definition: Left-hand Limit We say that L is the left-hand limit of f (x) at x 0 , written
lim x→x− 0
f (x) = L
if for every number > 0, there exists a corresponding number δ > 0 such that for all x x 0 − δ < x < x 0 =⇒ |f (x) − L| < .
We know the function f (x) = |x x| does not have a limit at x = 0, but it is reasonable to expect that it may have one sided limits at that point.
Example 1 Find limx→ 0 +^ f (x) and limx→ 0 −^ f (x) for f (x) = |x x|. Solution The solution to this problem becomes much more evident if we rewrite f (x) as
f (x) =
− 1 x < 0 1 x > 0
Now we can see that looking from just the left or right side of the point x = 0, we have two constant functions. Since the limit of a constant is just that constant, it follows that
lim x→ 0 +^
f (x) = 1 and lim x→ 0 −^
f (x) = − 1
It is useful to note that all of the properties of limits discussed earlier also hold for one-sided limits. The following theorem allows us to relate one-sided and two-sided limits.
Theorem: One-sided and Two-sided Limits A function f (x) has a limit L at x 0 if and only if it has right-hand and left-hand limits at x 0 , and both of those limits are L.
If we consider the definitions of right-hand and left-hand limits, we are creating a δ interval on one side of the point x 0. If limit is the same from both sides, then it stands to reason that we can we can create an interval around x 0 , which is exactly what we need to do to prove the limit exists. If the limit is different from both sides, then there is no limit at that point.
Infinite Limits
Let us consider the function f (x) = (^) x+2^1. If we look at the function as x → − 2 +^ we see that the function values increase without bound. For any B ∈ R we can choose x such that f (x) > B. Thus, limx→− 2 + f (x) does not exist. Even though the right-hand limit does not exist, we will write lim x→− 2 +^
f (x) = ∞
to describe that f (x) increases without bound, or approaches ∞ as x → − 2 +.
Similarly, If we look at the function as x → − 2 −^ we see that the function values decrease without bound. For any B ∈ R we can choose x such that f (x) < B. Thus, limx→− 2 − f (x) does not exist. Even though the left-hand limit does not exist, we will write
lim x→− 2 −^
f (x) = −∞
to describe that f (x) decreases without bound, or approaches −∞ as x → − 2 −. It is im- portant to stress that ∞ and −∞ are not numbers, so it is not possible for a function to have them as its limit. We only use this notation as a convenient way to say the function increases or decreases without bound in the limit.
Lastly, we describe two-sided limits using the exact same notation. If f (x) increases or decreases without bound as x approaches a point x 0 from both sides, then limx→x 0 f (x) = ±∞ respectively.
