1. Why One-Sided Limits Are Needed
Sometimes a function behaves differently when we approach a value from the left side and from the right side. To understand this behaviour, we look at one-sided limits — limits taken from only one direction.
This is especially useful for step functions, piecewise functions, or graphs with jumps.
2. Left-Hand Limit
The left-hand limit looks at the value a function approaches as x gets closer to a point from values less than that point.
2.1. Notation
\lim_{x \to a^-} f(x)
This reads as: “limit of f(x) as x approaches a from the left”.
2.2. Example
Consider the function:
f(x) = x + 1
To find the left-hand limit at x = 2:
\lim_{x \to 2^-} (x + 1) = 3
3. Right-Hand Limit
The right-hand limit looks at values approached when x comes from the right side, meaning values greater than the point.
3.1. Notation
\lim_{x \to a^+} f(x)
This reads as: “limit of f(x) as x approaches a from the right”.
3.2. Example
For the same function:
f(x) = x + 1
Right-hand limit at x = 2:
\lim_{x \to 2^+} (x + 1) = 3
4. When the One-Sided Limits Are Different
A limit exists at x = a only if both one-sided limits exist and are equal. When they differ, the limit does not exist, and the graph often has a jump or break.
4.1. Example of Unequal One-Sided Limits
Let:
f(x) = \begin{cases} 1, & x < 0 \\ 3, & x > 0 \end{cases}
Left-hand limit at 0:
\lim_{x \to 0^-} f(x) = 1
Right-hand limit at 0:
\lim_{x \to 0^+} f(x) = 3
Since 1 ≠ 3, the two-sided limit does not exist.
5. Graphical View of One-Sided Limits
One-sided limits are easier to see on a graph. Approaching from the left follows the graph from the left side; approaching from the right follows it from the right side. If both sides meet at the same height, the limit exists.
6. How One-Sided Limits Help
They help in understanding piecewise functions, dealing with jumps and sharp turns, and checking continuity at a point. They also are essential for defining derivatives.