Introduction to Limits
Understand what a limit means intuitively — approaching a value without necessarily reaching it — and learn why limits are the rigorous foundation beneath all of calculus.
Before calculus was placed on a rigorous foundation in the 19th century, mathematicians used derivatives and integrals without being able to fully justify them. The concept of the limit fixed this. A limit answers the question: "What value is this function getting closer and closer to, as the input approaches some number?" It's the mathematical language for "approaching" — and it underpins every derivative, every integral, and every continuity argument in calculus and analysis.
- Read limit notation and describe what a limit is asking
- Evaluate limits by substitution, factoring, or inspection of a table of values
- Identify cases where limits don't exist (oscillation, jump discontinuities, infinite limits)
"As x approaches a, f(x) approaches L." The limit is about the approach, not the arrival.
\[ \text{The limit exists} \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \]The function doesn't even need to be defined at \(x = a\) for the limit to exist there.
Approaching x = 2 for f(x) = (x² − 4)/(x − 2)
| x (from left) | f(x) | x (from right) | f(x) |
|---|---|---|---|
| 1.9 | 3.9 | 2.1 | 4.1 |
| 1.99 | 3.99 | 2.01 | 4.01 |
| 1.999 | 3.999 | 2.001 | 4.001 |
| 2 (undefined!) | 0/0 | Limit = 4 | |
Evaluate \(\lim_{x \to 3} (2x + 1)\).
The function is continuous at \(x = 3\), so just substitute:
\[ \lim_{x \to 3} (2x + 1) = 2(3) + 1 = \mathbf{7} \]Evaluate \(\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\).
Direct substitution gives 0/0 — an indeterminate form. Factor:
\[ \frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2 \quad \text{(for } x \neq 2\text{)} \] \[ \lim_{x \to 2} (x + 2) = \mathbf{4} \]A car's position is \(p(t) = t^2\) meters. Average speed between \(t = 3\) and \(t = 3 + h\):
\[ \frac{p(3+h) - p(3)}{h} = \frac{(3+h)^2 - 9}{h} = \frac{9 + 6h + h^2 - 9}{h} = \frac{6h + h^2}{h} = 6 + h \] \[ \lim_{h \to 0} (6 + h) = \mathbf{6 \text{ m/s}} \]This is the instantaneous speed at \(t = 3\) — and it's exactly how the derivative is defined.
- Evaluate \(\lim_{x \to 5} (x^2 - 3x + 1)\).
- Why can't you evaluate \(\lim_{x \to 1} \frac{x^2-1}{x-1}\) by direct substitution? What is the limit?
- If the left-hand limit is 7 and the right-hand limit is 7, does the limit exist?
▶ Show Answers
- \(25 - 15 + 1 =\) 11. (Direct substitution works since the function is continuous there.)
- Substituting \(x=1\) gives \(0/0\) (undefined). Factor: \((x-1)(x+1)/(x-1) = x+1\). Limit = \(1+1 =\) 2.
- Yes — when both one-sided limits agree, the two-sided limit exists and equals 7.
- Confusing the limit with the function value: The limit asks what f(x) approaches, not what f(x) equals at that point. The function might not even be defined at x = a.
- Thinking 0/0 means the limit is 0 or doesn't exist: 0/0 is indeterminate — it just means direct substitution failed. Factor, simplify, or use L'Hôpital's rule to find the actual limit.
- Forgetting to check both sides: A limit only exists if both the left-hand and right-hand limits agree. Always verify both sides for piecewise functions or near discontinuities.
- \(\lim_{x \to a} f(x) = L\) — read as "f(x) approaches L as x approaches a."
- Limits are about the approach — the function doesn't need to be defined at the target point.
- Direct substitution works when f is continuous. Use factoring for 0/0 forms.
- Limits are the formal foundation for derivatives (next lesson) and integrals.
Floating-point arithmetic in computers is essentially a discrete approximation of limits — as step sizes shrink toward zero, numerical methods (like Euler's method or Runge-Kutta) converge toward the true solution of differential equations. Physicists use limits constantly: Planck's constant is a limit as quantum effects approach the classical regime. Signal processing uses limits to define continuous Fourier transforms from discrete data. Any time computation approaches an exact answer as precision increases, you're working with a limit.
Calculator Connection
The Limit Calculator evaluates limits numerically and algebraically, showing the approach from both sides. The Function Plotter lets you visually inspect where a function has discontinuities or approaches a specific value.
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Introduction to Limits - Quiz
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