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Lesson 42 — Algebraic Properties of Limits

Limit of sum, product, quotient, power. Composition. Squeeze. Operational tools.

Used in: 2.º ano do EM (16-17 anos) · Equiv. Math II japonês (極限の性質) · Equiv. Oberstufe Grenzwertregeln alemão

\lim (f \pm g) = \lim f \pm \lim g, \quad \lim (fg) = \lim f \cdot \lim g, \quad \lim (f/g) = \lim f / \lim g

Choose your door

Rigorous notation, full derivation, hypotheses

Operational properties

Let $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ . Then:

Operation	Result
$\lim (f \pm g)$	$L \pm M$
$\lim (cf)$ , $c$ const.	$cL$
$\lim (fg)$	$LM$
$\lim (f/g)$ , $M \neq 0$	$L/M$
$\lim f^n$ , $n \in \mathbb{N}$	$L^n$
$\lim \sqrt[n]{f}$	$\sqrt[n]{L}$ (when defined)
$\lim	f

Composition

If $\lim_{x \to a} g(x) = b$ and $f$ is continuous at $b$ , then $\lim_{x \to a} f(g(x)) = f(b)$ .

Caution: without continuity, composition can fail. Counterexample: $g(x) = 0$ constant and $f$ with a removable discontinuity at 0.

Squeeze theorem (Sandwich)

If $g(x) \leq f(x) \leq h(x)$ in a neighborhood of $a$ (except possibly at $a$ ) and $\lim g = \lim h = L$ , then $\lim f = L$ .

Classic application

$\lim_{x \to 0} x \sin(1/x) = 0$ by squeeze: $-|x| \leq x \sin(1/x) \leq |x|$ and $\lim |x| = 0$ .

When it doesn't work

0/0, ∞/∞: indeterminate forms, require manipulation.
Denominator → 0: may give $\pm \infty$ or not exist.
Composition with discontinuous $f$ : requires care.

Exercise list

42 exercises · 10 with worked solution (25%)

Application 20Understanding 6Modeling 6Challenge 4Proof 6

Ex. 42.1Application
$\lim_{x \to 2} (x^2 + 3x - 1)$ . (Ans: 9.)
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Ex. 42.2ApplicationAnswer key
$\lim_{x \to 1} \sqrt{x^2 + 3}$ .
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Ex. 42.3ApplicationAnswer key
$\lim_{x \to 0} (x + 1)/(x^2 + 1)$ .
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Ex. 42.4ApplicationAnswer key
$\lim_{x \to 3} (x^2 - 9)/(x - 3)$ . (Ans: 6.)
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Ex. 42.5Application
$\lim_{x \to 0} (1 - \cos x)/x^2$ .
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Ex. 42.6Application
$\lim_{x \to 0} \sin(3x)/x$ . (Ans: 3.)
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Ex. 42.7Application
$\lim_{x \to 0} \tan x / x$ .
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Ex. 42.8Application
$\lim_{x \to \infty} (2x^2 - 3x + 1)/(x^2 + 5)$ . (Ans: 2.)
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Ex. 42.9Application
$\lim_{x \to \infty} (x + 1)/(x^2 + 1)$ .
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Ex. 42.10Application
$\lim_{x \to -\infty} (x^3 + 1)/(x^2 - 1)$ . (Ans: $-\infty$ .)
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Ex. 42.11Application
$\lim_{x \to 0} x^2 \sin(1/x)$ via squeeze.
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Ex. 42.12ApplicationAnswer key
$\lim_{x \to 0} (e^x - 1)/x$ .
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Ex. 42.13Application
$\lim_{x \to 0} \ln(1+x)/x$ .
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Ex. 42.14ApplicationAnswer key
$\lim_{x \to 1} \ln x / (x - 1)$ .
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Ex. 42.15Application
$\lim_{x \to 0} (1 + 2x)^{1/x}$ . (Ans: $e^2$ .)
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Ex. 42.16Application
$\lim_{x \to 4} (x - 4)/(\sqrt x - 2)$ .
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Ex. 42.17Application
$\lim_{x \to \infty} (\sin x)/x$ via squeeze.
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Ex. 42.18ApplicationAnswer key
$\lim_{n \to \infty} (2n^2 + 5n)/(n^2 + 1)$ .
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Ex. 42.19Application
$\lim_{x \to 0} (\cos x)^{1/x^2}$ . (Ans: $e^{-1/2}$ .)
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Ex. 42.20ApplicationAnswer key
$\lim_{x \to 0} (\sqrt{1+x} - \sqrt{1-x})/x$ . (Ans: 1.)
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Ex. 42.21Understanding
Show $\lim_{x \to 0} \sin x / x = 1$ by the squeeze theorem + unit-circle geometry.
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Ex. 42.22Understanding
Use squeeze to show $\lim_{x \to 0} x \cos(1/x) = 0$ .
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Ex. 42.23Understanding
Show that $\lim_{x \to 0^+} x^x = 1$ . (Use $x^x = e^{x \ln x}$ and $\lim x \ln x = 0$ .)
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Ex. 42.24Understanding
Give an example where $\lim (f + g)$ exists but $\lim f$ and $\lim g$ don't.
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Ex. 42.25Understanding
Give an example where $\lim (f \cdot g)$ exists but only one of $\lim f, \lim g$ exists.
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Ex. 42.26Understanding
Show that if $\lim |f(x)| = 0$ , then $\lim f(x) = 0$ .
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Ex. 42.27Proof
Prove $\lim (f + g) = \lim f + \lim g$ via ε-δ.
Ex. 42.28ProofAnswer key
Prove $\lim (cf) = c \lim f$ via ε-δ.
Ex. 42.29Proof
Prove the squeeze theorem via ε-δ.
Ex. 42.30Proof
Prove that if $\lim f = L$ and $L > 0$ , there exists a neighborhood where $f > L/2$ .
Ex. 42.31Proof
Prove that the limit (when it exists) is unique.
Ex. 42.32ProofAnswer key
Prove composition: $\lim_{x \to a} g = b$ , $f$ continuous at $b$ ⇒ $\lim f(g(x)) = f(b)$ .
Ex. 42.33Modeling
In control, transfer function $H(s) = (s+2)/(s^2 + 3s + 2)$ . Compute $H(0)$ via properties.
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Ex. 42.34ModelingAnswer key
In pharmacokinetics, $C(t) = (D/V)(e^{-k_a t} - e^{-k_e t})/(k_e - k_a)$ . Compute $\lim_{t \to \infty} C(t)$ .
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Ex. 42.35Modeling
Newton iteration sequence for $\sqrt 2$ : $a_{n+1} = (a_n + 2/a_n)/2$ . Use properties to find the limit.
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Ex. 42.36Modeling
Signal $V(t) = A \sin(\omega t)/t$ . Limit as $t \to \infty$ ? And $t \to 0$ ?
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Ex. 42.37Modeling
In probability, $X_n \to X$ in distribution. Show $E[\cos X_n] \to E[\cos X]$ via continuity of $\cos$ .
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Ex. 42.38Modeling
Euler method error: $|y(x_n) - y_n| \leq C h$ with $h \to 0$ . Model as a limit.
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Ex. 42.39Challenge
$\lim_{x \to 0} (\sin x - x)/x^3$ . (Ans: $-1/6$ .)
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Ex. 42.40Challenge
$\lim_{x \to 0} ((1+x)^{1/x} - e)/x$ .
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Ex. 42.41Challenge
$\lim_{x \to \infty} x \sin(\pi/x)$ . (Ans: $\pi$ .)
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Ex. 42.42Challenge
$\lim_{x \to 0} (\cos x - \cos(2x))/x^2$ . (Ans: $3/2$ .)
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Sources

Active Calculus — Boelkins · 2024 · §1.7-1.8.
Calculus (Volume 1) — OpenStax · 2016 · §2.3-2.4.
Basic Analysis — Lebl · 2024 · §3.1.