One-Sided Limits (Left-Hand and Right-Hand)

The two-sided limit lim_x→a f(x) exists only if both one-sided limits, lim_x→a⁻ f(x) (from the left) and lim_x→a⁺ f(x) (from the right), exist and are equal. When the two sides disagree, the two-sided limit does not exist (DNE). When they agree, the two-sided limit equals that common value. Evaluate one-sided limits the same way you evaluate two-sided limits: by direct substitution if the function is continuous on that side, by factoring or rationalizing if you get 0/0, or by checking the behavior of a piecewise or absolute value function near the point.

A left-hand limit is written lim_x→a⁻ f(x). The superscript minus sign means x approaches a from values less than a, from the left on the number line. A right-hand limit is written lim_x→a⁺ f(x). The superscript plus sign means x approaches a from values greater than a, from the right. In Stewart §2.2, these are defined formally with epsilon-delta: for the left-hand limit, you require a - δ < x < a; for the right-hand limit, a < x < a + δ. In practice, you never use epsilon-delta to compute a limit; you use algebraic techniques and the limit laws, which apply to one-sided limits identically as to two-sided limits (Stewart §2.3).

A left-hand limit considers only values of x that are less than a. For a function defined on both sides of a, check which rule applies as x comes in from the left. The failure case: if the function is not defined at all for x < a, as with √x at x = 0, the left-hand limit simply does not exist. The domain excludes it. For √x, lim_x→0⁺ √x = 0, but the left-hand limit is not defined.

A right-hand limit considers only values of x greater than a. For functions like 1/x at x = 0, the right-hand limit is ∞ because as x becomes a small positive number, 1/x grows without bound. The left-hand limit is -∞. Since these are not equal, the two-sided limit at 0 does not exist. The most common place students forget to check both sides: they see 1/x → ∞ from the right and assume the two-sided limit is ∞, but the left side goes to -∞.

A piecewise function is defined by different formulas on different intervals. At a point where the formula changes, use the left-hand piece to compute the left-hand limit and the right-hand piece to compute the right-hand limit. This is covered in Stewart §2.2. The two-sided limit exists only if the two pieces approach the same value.

Worked Example 1: A Piecewise Function

Let f(x) = { x² if x < 1; 2x if x ≥ 1 }. Find lim_x→1⁻ f(x) and lim_x→1⁺ f(x). For the left-hand limit at x = 1, use the piece x²: lim_x→1⁻ x² = 1. For the right-hand limit, use the piece 2x: lim_x→1⁺ 2x = 2. The left-hand limit is 1, the right-hand limit is 2. They are not equal, so the two-sided limit at x = 1 does not exist. This is a jump discontinuity.

Graph Illustration

Imagine the graph: the parabola x² ends at the point (1,1) with an open circle (since the piece x² is only for x < 1). The line 2x starts at (1,2) with a closed circle (since the piece 2x is defined at x = 1). There is a gap between y = 1 and y = 2 at x = 1. The function does not approach a single value from both sides, so the two-sided limit is DNE.

Absolute value functions create a corner point where the formula changes: |x| = x for x ≥ 0 and |x| = -x for x < 0. At x = 0, the left-hand limit uses -x and the right-hand uses x. For the classic example f(x) = |x|/x, Stewart §2.3 gives lim_x→0⁻ |x|/x = -1 and lim_x→0⁺ |x|/x = 1. The left-hand limit is -1, the right-hand limit is 1. They are not equal, so the two-sided limit at 0 does not exist.

Worked Example 2: Absolute Value

Find lim_x→0⁻ |x|/x and lim_x→0⁺ |x|/x. For x < 0, |x| = -x, so |x|/x = (-x)/x = -1. For x > 0, |x| = x, so |x|/x = x/x = 1. The left-hand limit is -1; the right-hand limit is 1. The two-sided limit does not exist. This is a jump discontinuity with a jump of size 2.

Graph Illustration

Graph f(x) = |x|/x. For all negative x, the function value is -1 (a horizontal line at y = -1, open circle at x = 0). For all positive x, the function value is 1 (a horizontal line at y = 1, open circle at x = 0). The function is not defined at x = 0. The left and right sides approach different y-values, so the two-sided limit is DNE.

When a function approaches ±∞ as x approaches a from one side, the vertical line x = a is a vertical asymptote (Stewart §2.2). The one-sided limit is written as ∞ or -∞, but remember: ∞ is not a real number, so the limit does not exist as a finite value. For 1/x at x = 0, lim_x→0⁻ 1/x = -∞ and lim_x→0⁺ 1/x = ∞. Since these disagree, the two-sided limit does not exist. For 1/x² at x = 0, both one-sided limits are ∞ (Stewart §2.2): lim_x→0⁻ 1/x² = ∞ and lim_x→0⁺ 1/x² = ∞. Here the one-sided limits agree (both are ∞), but the two-sided limit is still not a finite number; write lim_x→0 1/x² = ∞ to indicate the function grows without bound from both sides. The key distinction: for existence of a finite limit, the one-sided limits must be equal and finite. If they are equal but infinite, the limit does not exist as a real number.

Worked Example 3: Vertical Asymptote

Find the one-sided limits of f(x) = 1/(x-2) as x approaches 2. As x → 2⁻ (from the left), x - 2 is a small negative number, so 1/(x-2) → -∞. As x → 2⁺ (from the right), x - 2 is a small positive number, so 1/(x-2) → ∞. The left-hand limit is -∞, the right-hand limit is ∞. They are not equal, so the two-sided limit at x = 2 does not exist. The vertical line x = 2 is a vertical asymptote.

Graph Illustration

Graph f(x) = 1/(x-2). The curve has a vertical asymptote at x = 2. On the left of x = 2, the curve plunges down toward -∞. On the right of x = 2, the curve shoots up toward ∞. The two sides do not meet.

This rule (Stewart §2.2) is the central theorem of one-sided limits: lim_x→a f(x) = L if and only if lim_x→a⁻ f(x) = L and lim_x→a⁺ f(x) = L. Use one-sided limits to decide whether a two-sided limit exists. If the one-sided limits are different, the two-sided limit is DNE. If they are the same, the two-sided limit equals that common value. This applies to all limit problems, not just piecewise or absolute value functions. Any time the function behaves differently on the two sides of a point, whether due to a formula change, a vertical asymptote, oscillation (like sin(1/x) at 0), or a domain restriction, check both sides.

For the function (sin x)/x at x = 0, both one-sided limits equal 1 (Stewart §2.3). Since they agree, the two-sided limit exists and equals 1. For the greatest integer function floor(x) at an integer n, lim_x→n⁻ floor(x) = n - 1 and lim_x→n⁺ floor(x) = n (Stewart §2.3). They disagree, so the two-sided limit at each integer does not exist. This is a jump discontinuity.