One-Sided Limits: Everything You need to know

Evaluate a one-sided limit by approaching the target input from just the left or just the right. Compare both results to decide whether a two-sided limit exists.

What are One-Sided Limits?

Definition

A left-hand limit follows inputs smaller than the target; a right-hand limit follows inputs larger than it. The superscript signs identify the side of the input, not whether the output is positive or negative. The value at the target itself does not determine the limit.

Understanding the Notation

• \(\lim_{x \to c^+} f(x)\) signifies the limit of \(f(x)\) as \(x\) approaches \(c\) from the right.
• \(\lim_{x \to c^-} f(x)\) signifies the limit of \(f(x)\) as \(x\) approaches \(c\) from the left.

Why are One-Sided Limits Important?

• Analyzing functions that aren’t defined at a certain point.
• Understanding the behavior of functions near points of discontinuity.
• Studying the derivative of a function at a point.

How to Calculate One-Sided Limits

Step-by-Step Approach

1. Identify the Point of Interest: Determine the point \(c\) where you want to calculate the limit.
2. Approach from One Side: Decide whether you are calculating the limit from the left \(x \to c^-\) or from the right \(x \to c^+\).
3. Analyze Function Behavior: Examine how the function behaves as it gets closer to the point \(c\) from the chosen side.
4. Calculate the Limit: Use algebraic manipulation or graphical analysis to find the limit value as \(x\) approaches \(c\).

Examples

• Example 1: For \( f(x) = \frac {(x^2 – 4)}{(x – 2)}\), calculate \(\lim_{x \to 2^+} f(x)\) and \(\lim_{x \to 2^-} f(x)\).
• Example 2: For \( g(x) = \frac {|x|}{x}\), calculate \(\lim_{x \to 0^+} g(x)\) and \(\lim_{x \to 0^-} g(x)\).

Two worked one-sided limits

A hole does not prevent a limit

For the first example, factor the numerator: \(x^2-4=(x-2)(x+2)\). For every nearby input except 2, the fraction equals \(x+2\). From either side, its outputs approach 4. Thus \(\lim_{x\to2^-}f(x)=4\) and \(\lim_{x\to2^+}f(x)=4\). The two-sided limit is 4 even though the original fraction is undefined at 2. Cancellation does not define its missing value.

A jump gives different limits

For the second example, \(|x|=x\) when \(x>0\), so \(|x|/x=1\). When \(x<0\), \(|x|=-x\), so \(|x|/x=-1\). The right-hand limit is 1 and the left-hand limit is −1. Because they differ, the two-sided limit at 0 does not exist.

Compare the sides before concluding

At an interior point approached from both sides, a finite two-sided limit exists exactly when both one-sided limits have the same finite value. Check the branch used on each side for a piecewise function. For a graph, follow each branch toward the target; a filled dot shows the function value, which may differ from the approaching height.

For \(1/x\) near 0, the left side decreases without bound and the right side increases without bound. Write −∞ and +∞ to describe those behaviors; infinity is not a finite real limit. A few table entries can suggest a result but do not prove it.

Special Cases

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• Discontinuous Functions: The left-hand and right-hand limits might differ, indicating a discontinuity.
• Infinite Limits: Sometimes, one-sided limits approach infinity, showing an asymptotic behavior.

Quick self-check

Suppose a piecewise function equals x + 2 for x < 1 and 5 − x for x > 1, with any assigned value at x = 1. The left-hand limit is 3 and the right-hand limit is 4, so the two-sided limit does not exist. Changing only the value at 1 cannot make the approaching values agree.

FAQ

What is a one-sided limit?

A one-sided limit in calculus is the value a function approaches as its input gets closer to a certain point from one specific side, either from the left or the right.

How do you denote left-hand and right-hand limits?

The left-hand limit, as \(x\) approaches a point \(c\), is denoted as \(\lim_{x \to c^-} f(x)\), while the right-hand limit is denoted as \(\lim_{x \to c^+} f(x)\).

Why are one-sided limits important?

One-sided limits are important for understanding the behavior of functions at points of discontinuity, sharp turns, or where the function is not defined. They are essential in studying the properties of functions in a more detailed manner.

Can one-sided limits be different for the same point?

Yes, the left-hand and right-hand limits at the same point can be different, which usually indicates a discontinuity at that point.

How do you calculate one-sided limits?

To calculate a one-sided limit, you approach the point of interest from one side (left or right) and analyze how the function behaves as it gets closer to this point, using algebraic or graphical methods.

What does it mean if a one-sided limit is infinite?

If a one-sided limit approaches infinity, it suggests that the function grows without bound as it approaches the point from that specific side. This is indicative of an asymptotic behavior.

Are one-sided limits always finite?

No, one-sided limits can be finite, infinite, or might not exist, depending on the behavior of the function near the point of interest.

Can a function have a limit at a point if its one-sided limits are different?

No, for a function to have a limit at a point, both the left-hand and right-hand limits must exist and be equal at that point.

How do one-sided limits relate to derivatives?

One-sided limits are fundamental in defining the derivative of a function, especially at points where the function might not be continuous or differentiable in the traditional sense.

Where can I learn more about one-sided limits?

You can learn more about one-sided limits through calculus textbooks, online courses, tutorial videos, and interactive graphing tools that provide a visual understanding of these concepts.

For definitions and further examples, see OpenStax Calculus Volume 1: The Limit of a Function.