Limits & continuity

Learning goals

• Read and compute limits graphically, numerically, and algebraically.
• Use limit laws; resolve $0/0$ forms with algebra (conjugate, factor, rationalize).
• Define continuity; locate discontinuities and classify removable vs jump vs infinite.
• Apply the Intermediate Value Theorem in “root exists” style arguments.

Concept sheet

One-sided limits matter when the rule changes: $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$. A two-sided limit exists iff both one-sided limits exist and agree.

Continuity at $a$: $f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$. Practical checklist: (1) $f(a)$ defined, (2) $\lim_{x\to a}f(x)$ exists, (3) they are equal.

Limit laws: sums, products, quotients (denominator limit $\neq 0$), compositions behave as expected when the individual limits exist.

Worked patterns

Pattern A — polynomial / rational by substitution

If $f$ is rational and $a$ is in the domain (denominator $\neq 0$), then $\lim_{x\to a}f(x)=f(a)$.

Pattern B — removable $0/0$

Example: $\displaystyle \lim_{x\to 3}\frac{x^2-9}{x-3}=\lim_{x\to 3}\frac{(x-3)(x+3)}{x-3}=\lim_{x\to 3}(x+3)=6$.

Pattern C — squeeze intuition (template)

If $g(x)\le f(x)\le h(x)$ near $a$ and $\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L$, then $\lim_{x\to a}f(x)=L$.

Practice

Original drills — use Check when available, then Show solution. A quick celebration plays on correct auto-checks (respects reduced motion).

Homework / practice

WebAssign exports (MATH 151, Spring 2026). Order follows syllabus §2.2–2.6.

Section	Link	Focus
2.2 — Limit of a function	Open PDF	Graphs, tables, informal limits
2.3 — Limit laws	Open PDF	Algebra with limits
2.5 — Continuity	Open PDF	Continuity, IVT-style reasoning
2.6 — Limits at infinity	Open PDF	End behavior, horizontal asymptotes

Mistakes & checks

• $\lim_{x\to a} f(x)$ does not care about $f(a)$ if $f$ is undefined there — value is about nearby points.
• For piecewise functions at a break, always compare left vs right limits.
• IVT needs continuity on a closed interval, not “mostly continuous.”

Slide appendix (optional)

Optional condensed slide PDFs: export from your deck and drop under materials/; see materials/README.md.

Quiz mode

Homework PDF quiz: each card links to the same WebAssign export filenames as the semester map (materials/). A session shows 20 shuffled cards from 45 PDF-linked slots; exact problem wording stays in the PDF unless you merge transcripts per quiz/transcripts/README.md. Regenerate: python3 scripts/gen_hw_quiz.py.