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Lesson 45 — Fundamental Limits and Indeterminate Forms

lim sin x/x = 1, lim (1+1/n)^n = e, lim (e^x − 1)/x = 1. Indeterminate forms and techniques.

Used in: 2.º ano EM (Trim. 5) · Equiv. Math II japonês (cap. 3 — limites especiais) · Equiv. Klasse 11 alemã (Grenzwerte trigonometrisch) · Equiv. H2 Math singapurense (Special limits)

\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e, \quad \lim_{x \to 0} \frac{e^x - 1}{x} = 1

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Rigorous notation, full derivation, hypotheses

Fundamental limits and techniques

The 3 fundamentals

$\displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1$ (geometric — squeeze).
$\displaystyle\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$ (defines $e$ ).
$\displaystyle\lim_{x \to 0} \frac{e^x - 1}{x} = 1$ (derives from the previous).

Useful variants (Table)

Limit	Value
$\lim_{x \to 0} \sin x / x$	$1$
$\lim_{x \to 0} \sin(kx)/x$	$k$
$\lim_{x \to 0} \tan x / x$	$1$
$\lim_{x \to 0} (1-\cos x)/x^2$	$1/2$
$\lim_{x \to 0} (1-\cos x)/x$	$0$
$\lim_{x \to 0} \arcsin x / x$	$1$
$\lim_{x \to 0} \arctan x / x$	$1$
$\lim_{x \to 0} (e^x - 1)/x$	$1$
$\lim_{x \to 0} (a^x - 1)/x$	$\ln a$
$\lim_{x \to 0} \ln(1+x)/x$	$1$
$\lim_{x \to 0} ((1+x)^a - 1)/x$	$a$
$\lim_{x \to \infty} (1 + a/x)^x$	$e^a$
$\lim_{x \to 0^+} x^x$	$1$
$\lim_{x \to 0^+} x \ln x$	$0$

Common indeterminate forms (7)

$0/0$ , $\infty/\infty$ , $\infty - \infty$ , $0 \cdot \infty$ , $1^\infty$ , $0^0$ , $\infty^0$ — all solvable via manipulation or L'Hôpital (Lesson 64).

Techniques

Factor and cancel (polynomial $0/0$ ).
Conjugate ( $0/0$ with roots).
Trigonometric: use $\sin x \approx x$ (small $x$ ).
Variable change: $u = 1/x$ , $u = x - a$ .
Logarithm: $A^B = e^{B \ln A}$ converts $1^\infty$ , $0^0$ into $0/0$ .

Exercise list

42 exercises · 10 with worked solution (25%)

Application 30Understanding 4Modeling 6Challenge 2

Ex. 45.1Application
$\lim_{x \to 0} \sin(2x)/x$ . (Ans: 2.)
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Ex. 45.2Application
$\lim_{x \to 0} \sin(5x)/\sin(3x)$ . (Ans: $5/3$ .)
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Ex. 45.3Application
$\lim_{x \to 0} \tan x / x$ .
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Ex. 45.4Application
$\lim_{x \to 0} (1 - \cos x)/x^2$ . (Ans: $1/2$ .)
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Ex. 45.5Application
$\lim_{x \to \infty} (1 + 2/x)^x$ . (Ans: $e^2$ .)
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Ex. 45.6Application
$\lim_{x \to 0} (1 + 3x)^{1/x}$ .
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Ex. 45.7Application
$\lim_{x \to 0} (e^{2x} - 1)/x$ . (Ans: 2.)
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Ex. 45.8ApplicationAnswer key
$\lim_{x \to 0} (e^x - 1)/\sin x$ .
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Ex. 45.9Application
$\lim_{x \to 0} \ln(1+x)/x$ .
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Ex. 45.10ApplicationAnswer key
$\lim_{x \to 0} \ln(1 + 5x)/x$ .
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Ex. 45.11Application
$\lim_{x \to 0^+} x \ln x$ . (Ans: 0.)
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Ex. 45.12ApplicationAnswer key
$\lim_{x \to 0^+} x^x$ . (Ans: 1.)
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Ex. 45.13Application
$\lim_{x \to 1} (1/(1-x) - 2/(1-x^2))$ . (Ans: $-1/2$ .)
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Ex. 45.14Application
$\lim_{x \to 0} \sin(x^2)/x$ .
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Ex. 45.15ApplicationAnswer key
$\lim_{x \to \infty} (1 - 1/x)^x$ . (Ans: $1/e$ .)
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Ex. 45.16ApplicationAnswer key
$\lim_{x \to 0} ((1+x)^5 - 1)/x$ . (Ans: 5.)
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Ex. 45.17Application
$\lim_{x \to 0} (3^x - 1)/x$ . (Ans: $\ln 3$ .)
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Ex. 45.18Application
$\lim_{x \to 0} (\arcsin x)/x$ .
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Ex. 45.19Application
$\lim_{x \to 0} (\arctan x)/x$ .
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Ex. 45.20Application
$\lim_{x \to \infty} (x/(x+1))^x$ . (Ans: $1/e$ .)
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Ex. 45.21Application
$\lim_{x \to 0} (\cos x)^{1/x^2}$ . (Ans: $e^{-1/2}$ .)
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Ex. 45.22Application
$\lim_{x \to 0} (\cos x)^{1/x}$ . (Ans: 1.)
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Ex. 45.23ApplicationAnswer key
$\lim_{x \to 0} ((\sin x)/x)^{1/x^2}$ .
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Ex. 45.24Application
$\lim_{x \to \infty} ((x+2)/(x-1))^x$ . (Ans: $e^3$ .)
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Ex. 45.25ApplicationAnswer key
$\lim_{x \to 0^+} x^{\sin x}$ . (Ans: 1.)
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Ex. 45.26Application
$\lim_{x \to 0^+} (\sin x)^x$ . (Ans: 1.)
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Ex. 45.27Application
$\lim_{x \to 1} x^{1/(x-1)}$ .
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Ex. 45.28Application
$\lim_{x \to 0} (1 + \sin x)^{1/x}$ .
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Ex. 45.29Application
$\lim_{x \to 0} (e^x + x)^{1/x}$ .
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Ex. 45.30Application
$\lim_{n \to \infty} (n+1)/n^2 \cdot \sin(n)$ . (Ans: 0 via squeeze.)
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Ex. 45.31Modeling
Continuous compounding: $V_0 = 1000$ , $r = 5\%/$ year, $T = 10$ years. Compute $V(T)$ via $\lim_{n \to \infty} V_0(1 + r/n)^{nT}$ .
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Ex. 45.32Modeling
Radioactive decay: half-life 5 years. $N(t)/N_0$ after 12 years? Use $e^{-\lambda t}$ with $\lambda = \ln 2 / 5$ .
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Ex. 45.33ModelingAnswer key
Small-oscillation pendulum: $T = 2\pi\sqrt{L/g}$ . Justify substitution $\sin \theta \approx \theta$ via $\lim \sin x/x = 1$ .
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Ex. 45.34Modeling
Optical approximation: for small angles, $\sin \theta \approx \theta$ , $\tan \theta \approx \theta$ . Relative error at $\theta = 5°$ ?
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Ex. 45.35Modeling
Poisson distribution of rare events: $\lim_{n \to \infty, np = \lambda} \binom{n}{k} p^k (1-p)^{n-k} = e^{-\lambda}\lambda^k/k!$ .
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Ex. 45.36ModelingAnswer key
In finance, short-term American option: payoff approximated by $S_0 \cdot \sigma \sqrt T$ via expansion $\sin/(\cdot) \to 1$ .
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Ex. 45.37Understanding
Prove $\lim \sin x / x = 1$ via the unit circle and squeeze.
Ex. 45.38UnderstandingAnswer key
Show $\lim (1 + 1/x)^x$ is monotone increasing for $x \in \mathbb{N}$ .
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Ex. 45.39Understanding
Prove $\lim (e^x - 1)/x = 1$ via Taylor series.
Ex. 45.40Understanding
Show $\lim_{x \to 0} (a^x - 1)/x = \ln a$ .
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Ex. 45.41Challenge
$\lim_{x \to 0} (\tan x - \sin x)/x^3$ . (Ans: $1/2$ .)
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Ex. 45.42Challenge
$\lim_{x \to \infty} x(\ln(x+1) - \ln x)$ . (Ans: 1.)
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Sources

Active Calculus — Boelkins · 2024 · §1.8.
Calculus (Volume 1) — OpenStax · 2016 · §2.6.
APEX Calculus — Hartman · 2024 · §1.5.