Day 5 · Review + Quiz

Phase 1 Review

Every formula from Day 1–4 in one place. Skim it, then take the quiz below when you're ready.

Limit Notation & Algebraic Laws (Day 1)Limits at Infinity & Asymptotes (Day 2)Squeeze Theorem & Trig Limits (Day 3)Continuity & IVT (Day 4)

Limit Notation & Algebraic Laws (Day 1)

One-sided limits

\lim_{x \to c^-} f(x), \quad \lim_{x \to c^+} f(x)

The value $f(x)$ approaches from the left / right of $c$ .

Two-sided limit exists

\lim_{x \to c} f(x) \text{ exists} \iff \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)

Constant / identity

\lim_{x \to a} c = c, \quad \lim_{x \to a} x = a

Sum / difference / constant multiple

\lim_{x \to a} [f(x) \pm g(x)] = L \pm M, \quad \lim_{x \to a} [k \cdot f(x)] = k \cdot L

Product / quotient / power

\lim_{x \to a} [f(x) g(x)] = LM, \quad \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}, \quad \lim_{x \to a} [f(x)]^{\alpha} = L^{\alpha}

Quotient law requires $M \neq 0$ .

Direct substitution

\lim_{x \to a} p(x) = p(a)

For polynomials, and rational functions where $Q(a) \neq 0$ . If you get $0/0$ , factor or combine fractions first.

Limits at Infinity & Asymptotes (Day 2)

Horizontal asymptote by degree

\deg(N) < \deg(D)

y=0

.
If

\deg(N) = \deg(D)

y = \frac{\text{leading coef of } N}{\text{leading coef of } D}

.
If

\deg(N) > \deg(D)

: no HA.

Vertical asymptote

\lim_{x \to a} f(x) = \pm\infty

x=a

∞ − ∞ trick

Multiply by the conjugate, then re-apply degree comparison.

Used for $\sqrt{\cdot} - \sqrt{\cdot}$ or $\sqrt{\cdot} - x$ forms.

Squeeze Theorem & Trig Limits (Day 3)

Squeeze Theorem

g(x) \le f(x) \le h(x)

and

\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L \implies \lim_{x \to c} f(x) = L

Special trig limit #1

\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1

Special trig limit #2

\lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} = 0

Continuity & IVT (Day 4)

Continuity at a point

f(c)

defined,

\lim_{x \to c} f(x)

exists, and

\lim_{x \to c} f(x) = f(c)

Removable discontinuity

\lim_{x \to a} f(x)

exists but

\neq f(a)

(or

f(a)

undefined)

Jump discontinuity

\lim_{x \to a^+} f(x) \neq \lim_{x \to a^-} f(x)

Intermediate Value Theorem

f

continuous on

[a,b]

and

k

between

f(a), f(b) \implies \exists\, c \in (a,b): f(c) = k

Timed Multiple Choice Quiz

Phase 1 Quiz: Limits & Continuity

25 timed questions covering everything from Day 1–4. Treat it like a real exam — review your mistakes after.

45 min

Duration

Questions

Before you begin

·Pick one answer per question — you can change it any time before submitting.

·The timer starts immediately when you click Start.

·You can submit early at any time.

·Correct answers and explanations only appear after you submit.