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Unit 1 Cheat Sheet

Limits & Continuity — every formula, every decision rule, and every trap from the five lessons, on one printable page. Each block title links back to its lesson.
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What a Limit Is 1.1 – 1.4

A limit is the height the outputs approach from both sides — not the value the function takes there. Heading, not being.

two-sided exists when
average → instantaneous rate
Does the limit exist?
left limit = right limitlimit exists, equals that common value
one-sided limits differDNE (jump)
grows without boundDNE (infinite) — write \(\infty\) as shorthand
trap
  • The value at the point is irrelevant. \(\lim_{x\to a}f\) can exist even if \(f(a)\) is undefined or different.
  • Read the \(y\)-value, not the \(x\). A table marching toward \(x=1\) tells you nothing; watch what the outputs approach.

Computing Limits 1.5 – 1.7

Substitute first. If you get a number, you’re done. If you get \(\tfrac00\), do algebra to cancel the troublemaker.

factor & cancel
conjugate (radicals)
complex fraction
What did substitution give?
a numberthat’s the limit (function is continuous there)
\(\tfrac00\)factor, conjugate, or combine — then cancel and retry
\(\tfrac{k}{0}\), \(k\ne0\)infinite — limit DNE (vertical asymptote)
trap
  • \(\tfrac00\) is not an answer. It means "do more work," not "limit = 0" or "DNE."
  • \(\tfrac{k}{0}\) with \(k\ne0\) is different. That one really is DNE (infinite) — don’t try to cancel.

Squeeze Theorem 1.8 – 1.9

Trap a messy function between two friendly ones that meet. If both bounds \(\to L\), the middle is forced to \(L\) too.

the squeeze
the trig limits to memorize
Matching the sin / angle pattern
\(\tfrac{\sin kx}{x}\)\(=k\) — force the angle to match, compensate
\(\tfrac{\sin ax}{\sin bx}\)\(=\tfrac{a}{b}\)
bounded \(\times\) (something \(\to0\))\(\to 0\) by squeeze (e.g. \(x^2\cos\tfrac1x\))
trap
  • \(\tfrac{1-\cos x}{x}\to0\) but \(\tfrac{1-\cos x}{x^2}\to\tfrac12\). The power of \(x\) in the denominator decides.
  • Squeeze needs both bounds to meet. If they head to different values, it says nothing.

Continuity & Its Failures 1.10 – 1.13

Continuous means you can draw it without lifting your pen: where it’s heading equals where it is, with all three boxes checked.

the definition (all three at once)
solve-for-k at a seam
Classify the discontinuity
limit exists, value missing/wrongremovable (hole) — patchable
one-sided limits differjump — not patchable
blows up to \(\pm\infty\)infinite (asymptote) — not patchable
trap
  • Limit existing isn’t enough. You also need \(f(a)\) to exist AND equal the limit — all three.
  • Only removable breaks can be patched. Redefining one point can’t fix a jump or asymptote.
  • Solve-for-k: set the one-sided limits equal at the seam, then solve.
  • On an interval: continuous at every interior point; at a closed endpoint only the one existing side must match (e.g. \(\sqrt{x}\) is continuous on \([0,\infty)\)).

Asymptotes & the IVT 1.14 – 1.16

Limits at the extremes — vertical (blow-ups) and horizontal (end behavior) — then continuity’s payoff: the IVT.

horizontal asymptote (the degree race)
Intermediate Value Theorem
Horizontal asymptote: top vs bottom degree
bottom degree bigger\(y=0\)
equal degrees\(y=\) ratio of leading coefficients
top degree biggerno HA (grows without bound)
trap
  • Vertical asymptote needs num \(\ne0\) too. Denominator zero where numerator also zero is a hole, not an asymptote.
  • A graph can cross its horizontal asymptote. The HA only governs the far ends, not the middle.
  • IVT needs continuity on \([a,b]\). It proves a root exists — never where, never how many.

The Four Discontinuities quick ID

Every break is one of four creatures. Spot it by what the limit and value are doing.

Name that break
removablea single hole — limit exists, but \(f(a)\) is missing or wrong. Patchable.
jumpone-sided limits exist but disagree — the graph leaps. Fails the limit. Not patchable.
infiniteruns to \(\pm\infty\) at a vertical asymptote — denominator zero, numerator not. Not patchable.
oscillatingendless wiggles with no settling value, e.g. \(\sin\tfrac1x\) near \(0\). Fails the limit. Not patchable.
which box fails?
  • Removable fails box 1 or 3 (value missing or \(\ne\) limit) — the only fixable kind.
  • Jump, infinite, oscillating all fail box 2 (the limit itself doesn’t exist).
∞

Unit 1 is one question asked five ways: what is a function doing near a point? A limit names where it’s heading, algebra computes that value, the squeeze handles the wild cases, continuity asks whether heading equals being, and the IVT cashes that in — a continuous function skips no values. Everything in calculus stands on this.

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