Limits — Introduction to Calculus
Limit notation, the intuitive and formal definition, evaluating limits with four methods, one-sided limits, limits at infinity, and continuity — everything in Chapter 13.
What Is a Limit?
“As x approaches a, f(x) approaches L”
The limit may exist even if f(a) does not
If there is a hole in the graph at x = a, the limit can still exist. The limit asks about the approach, not the arrival.
The limit must agree from both sides
Both the left-hand limit (x→a⁻) and right-hand limit (x→a⁺) must exist and equal the same value L.
The limit and the function value can differ
Even if f(a) is defined, lim(x→a) f(x) might equal something different if there is a removable discontinuity.
Limit Laws
These laws hold when both lim f(x) and lim g(x) exist as x → a:
| Law | Formula |
|---|---|
| Sum | lim[f(x) + g(x)] = lim f(x) + lim g(x) |
| Difference | lim[f(x) − g(x)] = lim f(x) − lim g(x) |
| Constant multiple | lim[c · f(x)] = c · lim f(x) |
| Product | lim[f(x) · g(x)] = lim f(x) · lim g(x) |
| Quotient | lim[f(x)/g(x)] = lim f(x) / lim g(x), g(x) ≠ 0 |
| Power | lim[f(x)]ⁿ = [lim f(x)]ⁿ |
Four Methods for Evaluating Limits
Direct substitution
Works when f(a) is defined and continuous
- 1.Substitute x = a directly into f(x)
- 2.If the result is a real number, that is the limit
lim(x→3) (x² + 2x − 1) = 9 + 6 − 1 = 14
Factor and cancel
Works when direct substitution gives 0/0 (indeterminate form)
- 1.Factor the numerator and/or denominator
- 2.Cancel the common factor that causes the 0/0
- 3.Apply direct substitution to the simplified expression
lim(x→2) (x²−4)/(x−2) = lim (x+2)(x−2)/(x−2) = lim (x+2) = 4
Rationalize (conjugate)
Works when the expression contains a square root and gives 0/0
- 1.Multiply numerator and denominator by the conjugate of the radical expression
- 2.Simplify using (a−b)(a+b) = a²−b²
- 3.Cancel and apply direct substitution
lim(x→0) (√(x+4) − 2)/x → multiply by (√(x+4)+2)/(√(x+4)+2) → 1/(√4+2) = 1/4
Limit at infinity
Works as x → ±∞; divide by highest power of x
- 1.Identify the highest power of x in the expression
- 2.Divide every term by that power of x
- 3.Apply limits: as x→∞, 1/xⁿ → 0 for any positive n
lim(x→∞) (3x²+2x)/(x²−5) → divide by x²: (3+2/x)/(1−5/x²) → 3/1 = 3
One-Sided Limits
Left-hand limit
x approaches a from values less than a (from the left)
Right-hand limit
x approaches a from values greater than a (from the right)
When the limit does NOT exist
The two-sided limit exists only if both one-sided limits exist and are equal:
If left ≠ right, the limit does not exist (DNE). This occurs at jump discontinuities.
Limits at Infinity
When x → ∞ or x → −∞, we ask: what does f(x) approach as x grows without bound? The result describes horizontal asymptotes.
| Degree comparison (rational functions) | Limit as x → ∞ | Example |
|---|---|---|
| Numerator degree < denominator degree | 0 | lim (3x)/(x²+1) = 0 |
| Numerator degree = denominator degree | Ratio of leading coefficients | lim (3x²+2)/(x²−5) = 3 |
| Numerator degree > denominator degree | ±∞ (DNE as finite limit) | lim (x³)/(x²+1) = ∞ |
Continuity
A function f is continuous at x = a if all three conditions hold:
- 1f(a) is defined (no hole, no undefined value)
- 2lim(x→a) f(x) exists (left and right limits agree)
- 3lim(x→a) f(x) = f(a) (the limit equals the function value)
Removable discontinuity
A hole in the graph — f(a) is undefined, but the limit exists.
Can be 'filled' by defining f(a) = the limit
Jump discontinuity
Left and right limits exist but are not equal.
Cannot be fixed — the function must change definition
Infinite discontinuity
The function goes to ±∞ — vertical asymptote.
Cannot be removed — function is unbounded
Frequently Asked Questions
What is a limit in math?
A limit describes the value that a function approaches as the input approaches a certain value — but does not necessarily equal. Written lim(x→a) f(x) = L, this means: as x gets arbitrarily close to a (from either side), f(x) gets arbitrarily close to L. The limit may exist even if f(a) is undefined (like a hole in the graph) or if f(a) equals something different from L.
What is the difference between a limit and a function value?
A function value f(a) is what the function actually equals at x = a. A limit lim(x→a) f(x) is what the function approaches as x gets close to a. These are different. Example: f(x) = (x²−1)/(x−1) is undefined at x = 1 (0/0 form), but the limit as x→1 is 2, because (x²−1)/(x−1) = (x+1)(x−1)/(x−1) = x+1 → 2 as x → 1. The limit exists; the function value at x = 1 does not.
When does a limit not exist?
A limit does not exist (DNE) when: (1) the left-hand limit and right-hand limit are different (jump discontinuity), (2) the function oscillates infinitely near the point (like sin(1/x) as x→0), or (3) the function goes to ±∞ (vertical asymptote). For lim(x→a) f(x) to exist, the left limit lim(x→a⁻) and right limit lim(x→a⁺) must both exist and be equal.
Practice limit problems
All four evaluation methods, one-sided limits, continuity — with step-by-step solutions. Free to start, no account needed.
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