Sometimes a function behaves differently depending on which side you approach from. A piecewise function may follow one formula for x<a and a different formula for x>a. A rational function may blow up to +∞ on one side and −∞ on the other. In these situations, one-sided limits become essential.
The left-hand limit examines behavior as x approaches a through values less than a. The right-hand limit examines behavior through values greater than a. Each direction gets its own answer, and those answers need not agree.
One-sided limits serve as the building blocks for two-sided limits. The two-sided limit exists precisely when both one-sided limits exist and match. When they differ, the one-sided limits capture the full story that a single two-sided limit cannot tell.
Left-Hand Limits
The left-hand limit uses a minus superscript:
x→a−limf(x)
This notation means x approaches a through values strictly less than a. You move along the x-axis from the left, getting closer to a but never reaching or passing it.
Alternative notations include limx→a−f(x) and limx↑af(x). Some texts describe this as approaching "from below" since smaller x-values lie below a on the number line.
The left-hand limit asks: as x increases toward a, what value does f(x) approach?
Right-Hand Limits
The right-hand limit uses a plus superscript:
x→a+limf(x)
This notation means x approaches a through values strictly greater than a. You move along the x-axis from the right, getting closer to a but never reaching or passing it.
Alternative notations include limx→a+f(x) and limx↓af(x). Some texts describe this as approaching "from above" since larger x-values lie above a on the number line.
The right-hand limit asks: as x decreases toward a, what value does f(x) approach?
The Connection to Two-Sided Limits
The two-sided limit exists if and only if both one-sided limits exist and are equal:
x→alimf(x)=L⟺x→a−limf(x)=L and x→a+limf(x)=L
One-sided limits decompose the two-sided limit into its directional components. Checking whether a two-sided limit exists often begins with computing the one-sided limits separately.
If the one-sided limits exist but differ, the two-sided limit does not exist. If either one-sided limit fails to exist (due to oscillation or unbounded behavior), the two-sided limit also fails.
Piecewise Functions
Functions defined by different formulas on different intervals require one-sided limit analysis at the boundaries between pieces.
Consider:
f(x)={x23x−2x<2x≥2
At x=2, the left-hand limit uses the formula x2:
x→2−limx2=4
The right-hand limit uses the formula 3x−2:
x→2+lim(3x−2)=4
Since both one-sided limits equal 4, the two-sided limit exists and equals 4. If the formulas had produced different values, the two-sided limit would not exist.
Jump Discontinuities
A jump discontinuity occurs when both one-sided limits exist but differ. The function "jumps" from one value to another at the point.
The floor function ⌊x⌋ provides a standard example. At any integer n:
x→n−lim⌊x⌋=n−1
x→n+lim⌊x⌋=n
The left-hand limit gives the integer below, while the right-hand limit gives the integer itself. The function jumps by 1 at each integer. No two-sided limit exists at these points because the one-sided limits disagree.
One-Sided Limits at Vertical Asymptotes
Near a vertical asymptote, one-sided limits typically equal +∞ or −∞. The sign can differ depending on the direction of approach.
For f(x)=x−21:
x→2−limx−21=−∞
x→2+limx−21=+∞
From the left, x−2 is a small negative number, so the reciprocal is a large negative number. From the right, x−2 is a small positive number, so the reciprocal is a large positive number.
Expressions involving square roots often force one-sided analysis due to domain restrictions.
The expression 4−x requires 4−x≥0, meaning x≤4. At x=4, only the left-hand limit is meaningful:
x→4−lim4−x=0
Similarly, x−4 requires x≥4, so at x=4, only the right-hand limit applies:
x→4+limx−4=0
Domain boundaries naturally restrict limits to one side.
Evaluating One-Sided Limits
The same techniques used for two-sided limits apply: direct substitution, factoring, rationalizing. The difference lies in tracking which side you approach from.
Sign analysis becomes critical. For the expression x∣x∣:
x→0+limx∣x∣=xx=1
x→0−limx∣x∣=x−x=−1
When x>0, the absolute value ∣x∣=x. When x<0, the absolute value ∣x∣=−x. The direction of approach determines which case applies.
One-Sided Continuity
A function can be continuous from one side without being continuous from the other.
Continuous from the left at a:
x→a−limf(x)=f(a)
Continuous from the right at a:
x→a+limf(x)=f(a)
Full continuity at a requires both. On a closed interval [a,b], continuity means: continuous on the open interval (a,b), continuous from the right at a, and continuous from the left at b.