Welcome to Calculus I
Calculus is how we describe change: how a curve bends, how fast a quantity moves, and how small contributions add up to a total. Here you will move from limits and continuity into derivatives and their uses (including optimization), then into integration and the Fundamental Theorem—the bridge between instantaneous rate of change and accumulated growth.
Each chapter is built to be used, not skimmed: interact with live graphs, open worked examples line by line, and try practice problems with stepped solutions. Follow the numbered lessons in the sidebar at your own pace, take the Practice Exam when you are ready, and use Exam Remediation to focus on what still needs work.
2. Evaluate Basic Limits — direct substitution when $f$ is continuous at $a$
3. Solve Indeterminate Forms using Factoring — factor and cancel to resolve $\tfrac{0}{0}$
4. Rationalizing, and Simplifying Fractions — conjugate multiplication for radicals
5. Evaluating Limits at Infinity — divide by highest power, dominant terms
6. Special Limits & Theorems — Squeeze Theorem, trig limits ($\tfrac{\sin x}{x}$, $\tfrac{1-\cos x}{x^2}$), L'Hôpital
A limit describes the intended value of $f(x)$ as $x$ approaches $a$ — without $x$ ever needing to equal $a$. The value $f(a)$ may be undefined.
The limit exists only if both $\lim_{x\to a^-}f(x)$ and $\lim_{x\to a^+}f(x)$ exist and agree. For piecewise jumps, the left and right limits can differ.
When $\frac{0}{0}$ or $\frac{\infty}{\infty}$ arises, try: (1) factoring and canceling, (2) conjugate for radicals, (3) L'Hôpital's Rule: $\lim\frac{f}{g}=\lim\frac{f'}{g'}$.
The limit of $f(x)$ as $x$ approaches $a$ is the value $L$ that $f(x)$ gets arbitrarily close to — without $x$ ever needing to equal $a$.
$$\lim_{x \to a} f(x) = L$$
The value of $f(a)$ is irrelevant — $f$ need not even be defined at $a$.
$\lim[f\cdot g]=\lim f\cdot\lim g$
$\lim[f/g]=\lim f/\lim g\;(g\neq0)$
$\lim[f]^n=(\lim f)^n$
A limit exists if and only if both one-sided limits exist and agree:
$$\lim_{x\to a}f(x)=L \;\iff\; \lim_{x\to a^-}f(x)=L \;\text{ and }\; \lim_{x\to a^+}f(x)=L$$
1. Continuity
Why learn this?
Trust your graphs. Many formulas only make sense when a quantity changes smoothly—no sudden jumps where “just plug in” would give a misleading answer.
Helpful hints
Three-part checklist. Continuity at $a$ needs $f(a)$ defined, $\lim_{x\to a}f(x)$ to exist, and that limit to equal $f(a)$—if any link fails, the function is discontinuous there.
Remember
Removable gap when the limit exists but $f(a)$ is missing or wrong · One-sided mismatch means the two-sided limit does not exist
$f$ is continuous at $a$ when $f(a)$ is defined, $\displaystyle\lim_{x\to a}f(x)$ exists, and $\displaystyle\lim_{x\to a}f(x)=f(a)$.
- $f(a)$ is defined
- $\displaystyle\lim_{x\to a}f(x)$ exists
- $\displaystyle\lim_{x\to a}f(x)=f(a)$
In the Limit Visualizer above, the limit at $x=2$ is 4, but $f(2)$ is undefined — a removable discontinuity.
2. Evaluate Basic Limits
Why learn this?
Fast evaluation. A lot of limits come from “nice” functions built from polynomials, trig, exponentials, and logs—when things are continuous at the spot you care about, substitution is the first move.
Helpful hints
Check before you plug. Confirm $f$ is continuous at $a$ (polynomials and rationals where the denominator is nonzero are safe). If you get $\tfrac{0}{0}$ or other indeterminate forms, stop and simplify.
Remember
Continuous at $a$ $\Rightarrow$ $\lim_{x\to a}f(x)=f(a)$ · Composition of continuous functions is continuous on its domain
When $f$ is continuous at $a$, the limit equals $f(a)$. For polynomials, rationals (where defined), and compositions of continuous functions — just plug in.
3. Solve Indeterminate Forms using Factoring
Why learn this?
Rational expressions everywhere. Ratios of polynomials show up all over calculus; the limit can look like $\tfrac{0}{0}$ until you factor and cancel the piece that made both top and bottom zero.
Helpful hints
Factor first. Difference of squares, sum/difference of cubes, and grouping usually expose the $(x-a)$ factor that makes the numerator and denominator both vanish at $x=a$.
Remember
$\tfrac{0}{0}$ means “try algebra,” not “the limit is 0” · Cancel common factors valid for $x\to a$ because $x\neq a$ in the limit
When substitution yields $\tfrac{0}{0}$, factor and cancel the common factor. The limit of the simplified form is the same as the original.
4. Rationalizing, and Simplifying Fractions
Why learn this?
Radicals in measurements. Distances, norms, and wave speeds often involve square roots; rationalizing turns awkward $\sqrt{\cdot}$ differences into polynomials you can factor.
Helpful hints
Multiply by the conjugate. For $\sqrt{x}-c$, use $\sqrt{x}+c$ on top and bottom so $(\sqrt{x}-c)(\sqrt{x}+c)=x-c^2$ removes the radical from the numerator.
Remember
Conjugate flips the sign on the radical term · Simplify before substituting again
For limits with radicals (e.g. $\sqrt{x}-c$), multiply by the conjugate to remove the radical and reveal a cancelable factor.
5. Evaluating Limits at Infinity
Why learn this?
Long-run behavior. “What happens way out there?” is the same question as a limit at $\infty$: end behavior of a graph, long-term trends, horizontal asymptotes—not just one finite $x$.
Helpful hints
Divide by the highest power. Factor $x^n$ from numerator and denominator so every remaining term is a constant or tends to 0; then read off the horizontal asymptote or blow-up.
Remember
Equal degrees → ratio of leading coefficients · Denominator wins → limit $0$ · Numerator wins → $\pm\infty$ (watch signs)
Divide every term by the highest power of $x$. Terms with $x$ in the denominator vanish. The limit equals the ratio of leading coefficients when degrees match.
When the degrees don't match, the ratio of leading coefficients no longer applies:
- Denominator degree > numerator degree — After dividing by the highest power, the numerator vanishes while the denominator tends to a nonzero constant. The limit is 0. Example: $\displaystyle\lim_{x\to\infty}\frac{2x+1}{x^3-5}=\frac{0}{1}=0$.
- Numerator degree > denominator degree — The fraction behaves like a polynomial (e.g. $x^2$, $-x^3$) and grows without bound. The limit is $\pm\infty$, with the sign determined by the leading coefficients. Example: $\displaystyle\lim_{x\to\infty}\frac{5x^4-3x}{2x^2+1}\to+\infty$ (positive leading term dominates).
Summary: compare degrees first. If equal → ratio of leading coefficients. If denominator wins → 0. If numerator wins → $\pm\infty$.
6. Special Limits & Theorems
Why learn this?
Oscillating and messy ratios. The Squeeze Theorem and standard trig limits are the standard toolkit when algebra alone does not resolve the limit.
Helpful hints
Check the setup. L'Hôpital only applies when you truly have $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ (and the pieces are differentiable nearby). The Squeeze Theorem needs a lower and upper “fence” with the same limit.
Remember
$\lim_{x\to 0}\frac{\sin x}{x}=1$ · L'Hôpital differentiates numerator and denominator separately
Use when $f$ is trapped between two functions with the same limit — e.g. to prove $\lim_{x\to 0}\frac{\sin x}{x}=1$.
Use for $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ when factoring or algebraic tricks fail.
Fundamental limits used when trig appears in $\tfrac{0}{0}$ forms. Prove via Squeeze Theorem or L'Hôpital.