Limits, continuity, derivatives, and integrals — the big ideas of calculus explained from first principles, with the precalculus skills you need to succeed.
Calculus is often described as hard — but most students who struggle in calculus are actually struggling with the precalculus underneath it. The new ideas in Calc 1 are genuinely few; the algebra, trigonometry, and function fluency you bring in are what determine your success.
Every limit evaluation, every derivative, every integral involves simplifying expressions. Factoring, rational expressions, and exponent rules appear constantly. Weak algebra means stalling on mechanics rather than learning calculus concepts.
Calculus is about functions — their behavior near points, their rates of change, their accumulated values. If domain, range, composition, and inverse functions aren't solid, you can't reason about what calculus is doing.
Sine, cosine, the unit circle, and trig identities appear in derivatives, integrals, and applications immediately. Students who are shaky on precalculus trig hit a wall when calculus uses it as a starting assumption.
The foundation everything else is built on
The limit of f(x) as x approaches a is the value L that f(x) gets arbitrarily close to as x gets arbitrarily close to a — from either side:
The key insight: the limit is about what happens near x = a, not necessarily what happens at x = a. The function may be undefined at a, or may have a different value there, and the limit still exists.
x approaches a from the left (values less than a). The superscript minus sign indicates the direction of approach.
x approaches a from the right (values greater than a). The two-sided limit exists only when both one-sided limits agree.
If lim(x→a) f(x) = L and lim(x→a) g(x) = M, then:
| Law | Statement |
|---|---|
| Sum | lim [f(x) + g(x)] = L + M |
| Difference | lim [f(x) − g(x)] = L − M |
| Constant multiple | lim [c · f(x)] = c · L |
| Product | lim [f(x) · g(x)] = L · M |
| Quotient | lim [f(x) / g(x)] = L / M (M ≠ 0) |
| Power | lim [f(x)]ⁿ = Lⁿ |
| Root | lim ⁿ√f(x) = ⁿ√L (L ≥ 0 for even n) |
For polynomials and continuous functions, simply plug in x = a. If you get a real number, that is the limit.
When direct substitution gives 0/0, factor the numerator and/or denominator to cancel the offending factor, then substitute.
lim(x→2) (x² − 4)/(x − 2)
= lim(x→2) (x−2)(x+2)/(x−2)
= lim(x→2) (x + 2) = 4
Multiply numerator and denominator by the conjugate to eliminate radicals that are causing the indeterminate form.
lim(x→0) (√(x+4) − 2) / x
Multiply by (√(x+4) + 2) / (√(x+4) + 2):
= lim(x→0) (x+4 − 4) / [x(√(x+4) + 2)]
= lim(x→0) x / [x(√(x+4) + 2)]
= lim(x→0) 1 / (√(x+4) + 2) = 1/4
For rational functions as x → ±∞, compare the degrees of numerator and denominator.
deg(num) < deg(den)
Limit = 0
(3x)/(x² + 1) → 0
deg(num) = deg(den)
Limit = ratio of leading coefficients
(2x² + 1)/(5x² − 3) → 2/5
deg(num) > deg(den)
Limit = ±∞ (does not exist as a number)
(x³)/(x² + 1) → ±∞
f(a) is defined
The function must have a value at the point — no holes or undefined expressions.
lim(x→a) f(x) exists
The left-hand and right-hand limits must both exist and be equal.
lim(x→a) f(x) = f(a)
The limit must equal the function value — the graph does not jump away from or approach a different height than f(a).
A hole in the graph — the limit exists but f(a) is either undefined or has a different value. Can be 'filled in' by redefining f(a).
f(x) = (x²−1)/(x−1) at x = 1
The left-hand and right-hand limits both exist but are unequal. Common in piecewise functions. Cannot be made continuous by redefining a single point.
f(x) = x < 0: −1, x ≥ 0: 1
The function grows without bound near x = a — a vertical asymptote. The limit does not exist as a real number.
f(x) = 1/x at x = 0
Instantaneous rate of change and the tangent line
A secant line passes through two points on a curve: (x, f(x)) and (x+h, f(x+h)). Its slope is the average rate of change over the interval from x to x+h:
As h → 0, the second point slides along the curve toward the first. The secant line approaches the tangent line — a line that touches the curve at exactly one point and has the same slope as the curve at that instant. The slope of the tangent line is the derivative.
The derivative of f at x is defined as the limit of the difference quotient:
f'(x) = lim(h→0) [f(x+h) − f(x)] / h
Leibniz Notation
Emphasizes the ratio of infinitesimal changes. Useful in chain rule and related rates.
Prime Notation
Compact notation introduced by Newton. Common in single-variable calculus.
The derivative f'(a) is the instantaneous rate of change of f at x = a. Compare this to the average rate of change over an interval:
| Concept | Formula | Geometric Meaning |
|---|---|---|
| Average rate of change | [f(b) − f(a)] / (b − a) | Slope of secant line from a to b |
| Instantaneous rate of change | f'(a) = lim(h→0) [f(a+h)−f(a)]/h | Slope of tangent line at x = a |
f(x+h) = (x+h)² = x² + 2xh + h²
f(x+h) − f(x) = x² + 2xh + h² − x² = 2xh + h²
[f(x+h) − f(x)] / h = (2xh + h²) / h = 2x + h
lim(h→0) (2x + h) = 2x
f'(x) = 2x
Confirms the Power Rule: d/dx[xⁿ] = nxⁿ⁻¹. Here n = 2, so derivative = 2x¹ = 2x.
f(x+h) = 3(x+h) + 5 = 3x + 3h + 5
f(x+h) − f(x) = 3x + 3h + 5 − 3x − 5 = 3h
[f(x+h) − f(x)] / h = 3h / h = 3
lim(h→0) 3 = 3
f'(x) = 3
The derivative of a linear function is its slope. The constant 5 vanishes because it contributes zero rate of change.
From Example 1: f'(x) = 2x
At x = 3: f'(3) = 2(3) = 6
The point on the curve: (3, f(3)) = (3, 9)
Tangent line through (3, 9) with slope 6:
y − 9 = 6(x − 3) → y = 6x − 9
Tangent line: y = 6x − 9
Area under a curve and the Fundamental Theorem
The definite integral answers: what is the net signed area between f(x) and the x-axis from x = a to x = b? The idea is to approximate this area using rectangles (Riemann sums) and then take the limit as the rectangles get infinitely thin.
Δx = (b − a)/n is the width of each rectangle. As n → ∞, Δx → 0 and the sum approaches the exact area. Area above the x-axis is positive; area below is negative.
The FTC connects differentiation and integration — they are inverse processes. If F is an antiderivative of f (meaning F'(x) = f(x)), then:
Part I: Integration as Antidifferentiation
To compute ∫[a to b] f(x) dx, find any antiderivative F (such that F' = f), evaluate F(b) − F(a). No summing of rectangles needed.
Part II: The Accumulation Function
If G(x) = ∫[a to x] f(t) dt, then G'(x) = f(x). Differentiating an integral with a variable upper limit recovers the integrand.
| f(x) | ∫ f(x) dx |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹ / (n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
| k (constant) | kx + C |
C is the constant of integration — any constant has derivative 0, so antiderivatives are families of functions, not single functions.
If f is continuous on the closed interval [a, b] and N is any value strictly between f(a) and f(b), then there exists at least one c in the open interval (a, b) such that f(c) = N.
Intuition: A continuous function cannot skip values. If it starts at height f(a) and ends at height f(b), it must pass through every height in between.
Classic Application — Proving a Root Exists
Show f(x) = x³ − x − 1 has a root in (1, 2):
f(1) = 1 − 1 − 1 = −1 < 0
f(2) = 8 − 2 − 1 = 5 > 0
Since f is continuous and f(1) < 0 < f(2), by IVT there is a root in (1, 2). ✓
Use this checklist before starting Calc 1. Each gap here will slow you down once derivatives and integrals enter the picture.
A limit describes the value that a function approaches as its input gets arbitrarily close to a given number — not necessarily the value at that number. We write lim(x→a) f(x) = L to mean 'as x approaches a, f(x) gets closer and closer to L.' Limits are the foundation of all of calculus: derivatives and integrals are both defined as limits. You can often evaluate a limit by direct substitution, but when that produces an indeterminate form like 0/0, you need algebra (factoring, rationalizing) or L'Hôpital's Rule first.
The limit of f(x) as x→a is the value f approaches near a, while f(a) is the actual value at a. These can differ. For example, if f(x) = (x² − 1)/(x − 1), the function is undefined at x = 1, but the limit as x→1 equals 2 — because after factoring, (x−1)(x+1)/(x−1) = x+1, which approaches 2. A function can have a limit at a point even if it is undefined there. If f(a) exists and equals the limit, the function is called continuous at a.
A one-sided limit approaches from only one direction. The left-hand limit lim(x→a⁻) f(x) = L means x approaches a from values less than a. The right-hand limit lim(x→a⁺) f(x) = L means x approaches from values greater than a. The two-sided limit lim(x→a) f(x) exists if and only if both one-sided limits exist and are equal. One-sided limits are essential for piecewise functions and jump discontinuities — if the left and right limits differ, the overall limit does not exist at that point.
A function f is continuous at a point x = a if three conditions all hold: (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) lim(x→a) f(x) = f(a). Intuitively, you can draw the function through that point without lifting your pen. Polynomial and rational functions are continuous wherever they are defined. Common discontinuities include holes (removable), jumps, and vertical asymptotes (infinite discontinuities). Continuity on an interval means continuous at every point in that interval.
The derivative f'(a) measures the instantaneous rate of change of f at x = a — it is the slope of the tangent line to the graph at that point. Formally, f'(a) = lim(h→0) [f(a+h) − f(a)] / h. This expression is the difference quotient, and its limit as h approaches 0 gives the exact slope at a single point rather than the average slope between two points. Derivatives answer questions like 'how fast is this quantity changing right now?' in physics (velocity), economics (marginal cost), and biology (growth rate).
The difference quotient [f(x+h) − f(x)] / h gives the average rate of change of f over the interval from x to x+h — it is the slope of the secant line through (x, f(x)) and (x+h, f(x+h)). As h shrinks toward 0, the secant line rotates until it becomes the tangent line at x, and the difference quotient approaches the derivative f'(x). So the derivative is defined as the limit of the difference quotient: f'(x) = lim(h→0) [f(x+h) − f(x)] / h. Every derivative rule (power rule, chain rule) is derived from this definition.
The definite integral ∫[a to b] f(x) dx measures the net signed area between the graph of f and the x-axis from x = a to x = b. Area above the axis is positive; area below is negative. The integral is defined as the limit of Riemann sums — you slice the region into thin rectangles of width Δx, sum their areas, and let Δx approach 0. The Fundamental Theorem of Calculus connects derivatives and integrals: if F is an antiderivative of f (meaning F' = f), then ∫[a to b] f(x) dx = F(b) − F(a). Integration undoes differentiation.
The most critical precalculus skills for calculus success are: (1) Algebra fluency — factoring polynomials, simplifying rational expressions, solving equations. (2) Function concepts — domain, range, composition, inverses. (3) Graphing — recognizing and sketching lines, polynomials, rational functions, exponentials, logarithms, and trig functions. (4) Trigonometry — unit circle values, identities (sin² + cos² = 1, double-angle), and trig equations. (5) Exponent and logarithm rules. (6) Equation of a line (slope, slope-intercept, point-slope). Students who struggle in calculus almost always have gaps in one of these areas.
The Intermediate Value Theorem (IVT) states: if f is continuous on the closed interval [a, b] and N is any value between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N. In plain English: a continuous function cannot jump over a value — it must pass through every value between its endpoints. A classic application is proving that a root exists: if f(a) < 0 and f(b) > 0 and f is continuous, there must be some c where f(c) = 0.
Many calculus problems involve the form 0/0 or ∞/∞ when you plug in directly, which are indeterminate — they have no single determined value. The slope of a tangent line, for instance, requires dividing rise by run as the run approaches zero: 0/0. Limits provide a rigorous way to handle this by asking 'what value does the expression approach?' rather than 'what is the expression at that point?' Substitution works only when the function is continuous at the point; limits handle the cases where it is not, making them the correct foundation for all of calculus.
Epsilon-delta definition, squeeze theorem, and limit strategies for all function types
Domain, range, composition, inverses, and transformations — the objects calculus studies
Pythagorean, double-angle, sum and difference formulas — essential for calculus integration
Interactive limit and derivative problems with step-by-step AI explanations — free to try.
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