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Lesson 50 — Term 5 Consolidation: Limits and Continuity

Integration workshop. ε-δ limits, properties, fundamentals, continuity, IVT, asymptotes, sequences.

Used in: 2.º ano EM (16-17 anos) · Equiv. Analysis I (Gymnasium alemão) · Equiv. Math II japonês — seção limites

\lim_{x \to a} f(x) = L \iff \forall \varepsilon > 0, \exists \delta > 0 : 0 < |x-a| < \delta \Rightarrow |f(x) - L| < \varepsilon

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Rigorous notation, full derivation, hypotheses

Term 5 synthesis and map

You have completed the rigorous calculus foundations. A map of what was covered:

Lesson	Topic	Key concept
41	Formal limit	$\eps$ - $\delta$
42	Properties	Sum, product, quotient, squeeze
43	Continuity	$\lim f = f(a)$ , types of discontinuity
44	One-sided and infinite	Existence via one-sided
45	Fundamentals	$\sin x/x$ , $(1+1/n)^n$ , $(e^x-1)/x$
46	IVT	Existence of roots, bisection
47	Asymptotes	VA, HA, OA
48	Trig	Manipulating $\sin, \cos, \tan$
49	Sequences	Cauchy, Bolzano-Weierstrass

Theorems summary table

Theorem	Hypothesis	Conclusion
Squeeze	$g \leq f \leq h$ , $\lim g = \lim h = L$	$\lim f = L$
IVT	$f \in C([a,b])$ , $k$ between $f(a), f(b)$	$\exists c : f(c) = k$
Weierstrass	$f \in C([a,b])$	$f$ attains max and min
Bolzano-Weierstrass	bounded $(a_n)$	Has convergent subsequence
Heine-Cantor	$f \in C([a,b])$	$f$ uniformly continuous
Cauchy	$(a_n)$ Cauchy in $\mathbb{R}$	Converges
Monotone bounded	increasing, bounded above	Converges

Next step

Derivatives (Term 6) — defined as a limit: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$

All Term 5 fluency in limits will be used directly there.

Manipulation cheat sheet

Form	Technique
Polynomial $0/0$	Factor and cancel
$0/0$ with roots	Conjugate
Trig $0/0$	Fundamental limits
Rational $\infty/\infty$	Divide by highest degree
$1^\infty$	$A^B = e^{B \ln A}$
$0 \cdot \infty$	Rewrite as quotient
$\infty - \infty$	Common factor, conjugate

Exercise list

40 exercises · 10 with worked solution (25%)

Application 22Understanding 3Modeling 8Challenge 4Proof 3

Ex. 50.1Application
$\lim_{x \to 2} (3x^2 - x + 1)$ . (Ans: 11.)
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Ex. 50.2Application
$\lim_{x \to 1} (x^2 - 1)/(x - 1)$ . (Ans: 2.)
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Ex. 50.3Application
$\lim_{x \to 0} \sin(5x)/x$ . (Ans: 5.)
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Ex. 50.4Application
$\lim_{x \to 0} (1 - \cos x)/x^2$ . (Ans: $1/2$ .)
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Ex. 50.5Application
$\lim_{x \to \infty} (1 + 3/x)^x$ . (Ans: $e^3$ .)
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Ex. 50.6Application
$\lim_{x \to \infty} (2x^2 + x)/(x^2 - 5)$ . (Ans: 2.)
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Ex. 50.7Application
$\lim_{x \to 0^+} x \ln x$ . (Ans: 0.)
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Ex. 50.8Application
Asymptotes of $f(x) = (x^2 + 1)/(x - 2)$ . (Ans: VA $x = 2$ , OA $y = x + 2$ .)
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Ex. 50.9Application
Is $f(x) = (x^2 - 9)/(x - 3)$ continuous at $x = 3$ ? Fix it. (Ans: $f(3) = 6$ .)
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Ex. 50.10Application
$\lim_{x \to 0} (e^{2x} - 1)/(\sin(3x))$ . (Ans: $2/3$ .)
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Ex. 50.11Application
$\lim n!/n^n$ . (Ans: 0.)
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Ex. 50.12ApplicationAnswer key
Find $a$ such that $f(x) = \begin{cases} x + a & x < 0 \\ x^2 + 1 & x \geq 0 \end{cases}$ is continuous. (Ans: $a = 1$ .)
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Ex. 50.13Application
$\lim_{x \to 0} (\tan x - x)/x^3$ . (Ans: $1/3$ .)
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Ex. 50.14ApplicationAnswer key
$\lim_{x \to 0} (1 - \cos(2x))/(x^2)$ .
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Ex. 50.15Application
$\lim_{x \to \infty} \sqrt{x^2 + x} - x$ . (Ans: $1/2$ .)
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Ex. 50.16Application
$\lim_{n \to \infty} (1 - 2/n)^n$ . (Ans: $e^{-2}$ .)
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Ex. 50.17Application
$\lim_{x \to 0} \ln(1+5x)/x$ . (Ans: 5.)
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Ex. 50.18Application
Asymptotes of $f(x) = \arctan x + 1/x$ . (Ans: VA $x=0$ , HAs $y = \pm \pi/2$ .)
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Ex. 50.19Application
$\lim_{x \to \pi/2^-} \tan x \cdot (\pi/2 - x)$ . (Ans: 1.)
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Ex. 50.20ApplicationAnswer key
$\lim_{n \to \infty} \sum_{k=1}^n 1/n \cdot 1/(1 + (k/n)^2)$ . (Ans: $\pi/4$ via integral.)
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Ex. 50.21Application
$f(x) = \begin{cases} \sin x / x & x \neq 0 \\ 1 & x = 0 \end{cases}$ . Continuous at 0? (Ans: Yes.)
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Ex. 50.22ModelingAnswer key
Does $x^3 - 2x - 1 = 0$ have a root in $(1, 2)$ ? IVT. (Ans: Yes.)
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Ex. 50.23Modeling
Show that $\cos x = x^2$ has a solution in $(0, 1)$ .
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Ex. 50.24Modeling
Does $f(x) = e^x - x - 2$ have a root in $\mathbb{R}^+$ ? Where?
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Ex. 50.25Application
Where is $f(x) = (x^2 - 4)/(x^2 - 5x + 6)$ discontinuous? Types?
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Ex. 50.26ModelingAnswer key
In RC: $V(t) = V_\infty (1 - e^{-t/RC})$ . $\lim_{t \to \infty} V(t) = V_\infty$ . Verify. Time for $99\%$ of $V_\infty$ ?
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Ex. 50.27ModelingAnswer key
In radioactive decay, $N(t) = N_0 e^{-t/\tau}$ . Half-life in terms of $\tau$ . Limit as $t \to \infty$ ?
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Ex. 50.28ModelingAnswer key
In control, $H(s) = (s+1)/(s^2 + 4s + 5)$ . Compute $H(0)$ , $\lim_{|s| \to \infty} H(s)$ .
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Ex. 50.29Modeling
In finance, European option $C(S, T) \to S$ as $S \to \infty$ . Confirm via limit on the Black-Scholes formula.
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Ex. 50.30Modeling
In optimization, gradient descent $w_{n+1} = w_n - \alpha \nabla f(w_n)$ converges if $f$ is convex $L$ -Lipschitz and $\alpha < 2/L$ . Show via sequence analysis.
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Ex. 50.31UnderstandingAnswer key
Construct $f$ discontinuous everywhere (Dirichlet) and justify.
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Ex. 50.32UnderstandingAnswer key
Show via Bolzano-Weierstrass that $\sin n$ has a convergent subsequence.
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Ex. 50.33Understanding
Show that if $f$ is continuous on $[a, b]$ , it is uniformly continuous (Heine-Cantor).
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Ex. 50.34Proof
Prove $\lim_{x \to a} f(x) = L$ if and only if both one-sided limits exist and equal $L$ .
Ex. 50.35Proof
Prove that $f \in C([a,b])$ is bounded.
Ex. 50.36Proof
Prove that an increasing bounded sequence is Cauchy.
Ex. 50.37ChallengeAnswer key
$\lim_{x \to 0} (\sin x)^{1/x}$ . Does it exist? Compute one-sided.
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Ex. 50.38Challenge
$\lim_{n \to \infty} (n!)^{1/n^2}$ .
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Ex. 50.39Challenge
$f(x) = x \cos(1/x)$ if $x \neq 0$ , $f(0) = 0$ . Show that $f$ is continuous at 0 but not differentiable.
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Ex. 50.40Challenge
$\lim_{x \to \infty} \ln(\Gamma(x))/(x \ln x)$ via Stirling. (Ans: 1.)
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Sources

Active Calculus — Boelkins · 2024 · §1.7-1.9.
Calculus (Volume 1) — OpenStax · 2016 · ch. 2.
Basic Analysis — Lebl · 2024 · §3.
APEX Calculus — Hartman · 2024 · ch. 1.