limits rq review

What is a limit point, and what is a punctured neighbourhood?

a is a limit point of A iff every Bδ(a)\{a} meets A; Bδ(a)\{a}={x:0<|x−a|<δ}.

Formal ε–δ definition of limₓ→ₐ f(x)=L ∈ ℝ.

∀ε>0 ∃δ>0 such that ∀x∈A: 0<|x−a|<δ ⇒ |f(x)−L|<ε. The point a is a limit point of A.

General neighbourhood definition of limₓ→ₐ f(x)=L?

∀ neighbourhoods V(L), ∃ neighbourhood U(a): x∈(U(a)\{a})∩A ⇒ f(x)∈V(L). Applies also to ±∞.

Define an infinite limit as x→a, with a finite.

limₓ→ₐ f(x)=+∞ iff ∀M>0 ∃δ>0: x∈A, 0<|x−a|<δ ⇒ f(x)>M. For −∞ use f(x)<−M.

Define limₓ→+∞ f(x)=L∈ℝ (and the −∞ version).

∀ε>0 ∃M>0: x∈A, x>M ⇒ |f(x)−L|<ε. At −∞ replace x>M with x<−M.

Define right- and left-hand finite limits at a.

Right: ∀ε>0 ∃δ>0: x∈A, a<x<a+δ ⇒ |f(x)−L|<ε. Left: a−δ<x<a. a must be approachable from that side.

When does a two-sided limit at a equal L?

Provided both sides of a are approachable from A: limₓ→ₐ f(x)=L ⇔ limₓ→ₐ⁻ f(x)=limₓ→ₐ⁺ f(x)=L (finite or extended).

Difference: limₓ→ₐ⁺f(x)=L vs limₓ→ₐ f(x)=L⁺?

a⁺ describes x approaching a FROM THE RIGHT; L⁺ describes f(x) approaching L FROM ABOVE (eventually f(x)>L).

Define vertical and horizontal asymptotes.

Vertical x=a: at least one one-sided limit at a equals ±∞. Horizontal y=L: limₓ→+∞f(x)=L or limₓ→−∞f(x)=L, L finite.

Sequential characterization of a function limit.

limₓ→ₐ f(x)=L ⇔ for every sequence xₙ∈A\{a} with xₙ→a, we have f(xₙ)→L. Valid for scalar or vector inputs.

Uniqueness of finite limits (H): statement and proof idea?

A function limit at a, if finite, is unique. If L≠M were limits, choose ε<|L−M|/2; near a, f(x) cannot be within ε of both.

Sign permanence theorem for finite nonzero limit.

If limₓ→ₐ f(x)=L≠0, then f(x) has the same sign as L throughout some punctured neighbourhood of a.

State the comparison/squeeze theorem (H).

If g(x)≤f(x)≤h(x) near a and lim g=lim h=L (finite), then lim f=L. Proof: combine bounds within ε using minimum δ.

Sum-of-function-limits theorem (*), finite case?

If f(x)→L∈ℝ and g(x)→M∈ℝ as x→a, then (f+g)(x)→L+M.

Prove finite limit of sum (*), via ε–δ.

Choose δf for ε/2 and δg for ε/2; δ=min(δf,δg). Then |(f+g)−(L+M)|≤|f−L|+|g−M|<ε.

Limit algebra: product and quotient of functions?

If f→L,g→M finite, then fg→LM. If M≠0 and g≠0 near a, f/g→L/M.

Why does the quotient rule need a nonzero denominator limit?

If g(x)→0, quotient f/g may be indeterminate or diverge; the finite quotient theorem requires lim g=M≠0.

What are the main indeterminate forms for limits?

0/0, ∞/∞, ∞−∞, 0·∞; also 0⁰, 1^∞ and ∞⁰ for powers. They require further analysis.

Fundamental sine limit as x→0?

sin x/x→1 (x measured in radians); equivalently sin x∼x.

Fundamental cosine limit as x→0?

(1−cos x)/x²→1/2; equivalently 1−cos x∼x²/2.

Fundamental exponential limit as x→0?

(eˣ−1)/x→1; equivalently eˣ−1∼x.

Fundamental natural-log limit as x→0?

ln(1+x)/x→1; equivalently ln(1+x)∼x.

Fundamental power limit as x→0?

[(1+x)^α−1]/x→α (fixed α∈ℝ, where defined); for α≠0, (1+x)^α−1∼αx.

Fundamental limit defining e (and generalisation)?

(1+1/t)^t→e as t→+∞; more generally (1+k/t)^t→e^k for fixed k.

Define f=o(g) as x→a.

f=o(g) iff f(x)/g(x)→0 as x→a, with g(x)≠0 near a. f is negligible relative to g.

Define asymptotic equivalence f∼g as x→a.

f∼g iff f(x)/g(x)→1 as x→a, with g(x)≠0 near a.

Characterise asymptotic equivalence via little-o.

f∼g ⇔ f=g+o(g), with g nonzero near a. This is the basic error representation.

Can you replace f∼g in a difference?

Not generally: leading terms can cancel. Equivalences work safely for products/quotients, not arbitrary sums or differences.

Growth hierarchy as x→+∞?

For p>0, a>1: ln x=o(xᵖ) and xᵖ=o(aˣ). Logarithms grow slower than powers, which grow slower than exponentials.

Define continuity at a, including isolated points.

At a∈A that is a limit point: limₓ→ₐ f(x)=f(a). At an isolated point of A, f is continuous automatically.

ε–δ definition of continuity at a∈A.

∀ε>0 ∃δ>0: x∈A and ‖x−a‖<δ ⇒ |f(x)−f(a)|<ε.

Sequential criterion for continuity at a.

f is continuous at a iff every sequence xₙ∈A with xₙ→a satisfies f(xₙ)→f(a).

Define a removable discontinuity.

Finite limₓ→ₐ f(x)=L exists, but f(a)≠L or f(a) is undefined. Redefining f(a)=L repairs it.

Define a jump discontinuity.

Finite left- and right-hand limits exist at a but are unequal; changing f(a) cannot fix it.

Define an essential discontinuity.

At least one one-sided limit at a is infinite or fails to exist; it cannot be repaired merely by redefining f(a).

Which operations preserve continuity at a?

If f,g continuous: f+g and fg continuous; f/g if g(a)≠0; g∘f if g continuous at f(a).

State Bolzano's zero-value theorem.

If f is continuous on [a,b] and f(a)f(b)≤0, then ∃c∈[a,b] with f(c)=0. Opposite strict signs give c∈(a,b).

State Weierstrass's theorem.

If C⊆ℝⁿ is nonempty and compact (closed and bounded) and f:C→ℝ is continuous, f attains a global min and max.

State the intermediate-value theorem (Darboux).

If f is continuous on [a,b], it takes every value between f(a) and f(b); indeed f([a,b])=[min f,max f].

Continuous f on interval I: injectivity vs monotonicity?

f is injective ⇔ f is strictly monotone on I. It has inverse f⁻¹:f(I)→I, strictly monotone in the same direction.

ε–δ definition of a multivariable function limit.

For a limit point a∈ℝⁿ: ∀ε>0 ∃δ>0: x∈A, 0<‖x−a‖<δ ⇒ |f(x)−L|<ε.

How to disprove or prove a 2-variable limit using paths?

Two paths to a with different limits DISPROVE it. Agreement along lines alone does NOT prove it; a curved path may differ.

Define coercivity of f on C (textbook definition).

∃t∈ℝ such that upper contour set {x∈C:f(x)≥t} is NONEMPTY and COMPACT. Only one such level is required.

Define supercoercivity; relation to coercivity.

For every sequence xₙ with ‖xₙ‖→∞, f(xₙ)→−∞. With continuity on closed C, supercoercivity ⇒ coercivity; converse fails.

State Tonelli's theorem (*).

If f:C→ℝ is continuous and coercive on C, then f attains a global MAXIMUM on C (not necessarily a minimum).

Proof idea for Tonelli's theorem (*).

Pick nonempty compact K={x∈C:f(x)≥t}; Weierstrass gives max on K. Outside K, f<t≤max_K f, so it is global on C.

General-base exponential fundamental limit as x→0?

lim (aˣ−1)/x = ln a for a>0 (a≠1 for a nontrivial exponential).

General-base logarithm fundamental limit as x→0?

lim logₐ(1+x)/x = 1/ln a, for a>0 and a≠1.

Fundamental e-limit with x→0?

lim (1+x)^(1/x) = e as x→0, with x≠0 and 1+x>0.

Fundamental first-order cosine limit as x→0?

lim (1−cos x)/x = 0. Also lim (1−cos x)/x² = 1/2.