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Lesson 47 — Asymptotes and Asymptotic Behavior

Vertical, horizontal, oblique asymptotes. Sketching graphs via asymptotic analysis.

Used in: 2.º ano EM · Equiv. Math II japonês cap. 5 · Equiv. Klasse 11 alemã análise de funções

\text{VA: } x = a, \quad \text{HA: } y = L, \quad \text{OA: } y = mx + b

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Rigorous notation, full derivation, hypotheses

Asymptotes — definitions and computation

Vertical asymptote (VA)

Where they appear: points where $f$ has a pole (zero denominator, log of 0, etc.).

Horizontal asymptote (HA)

There may be two HAs (one at $+\infty$ , another at $-\infty$ ): for example $\arctan x$ has $y = \pi/2$ and $y = -\pi/2$ .

Oblique asymptote (OA)

Computation: $m = \lim_{x \to \pm\infty} \frac{f(x)}{x}, \quad b = \lim_{x \to \pm\infty} (f(x) - mx).$

If both limits exist with finite non-zero $m$ , there is an OA.

Asymptote table for $P(x)/Q(x)$

Case	Asymptotes
$\deg P < \deg Q$	HA at $y = 0$
$\deg P = \deg Q$	HA at $y =$ ratio of leading coeffs
$\deg P = \deg Q + 1$	OA ( $y =$ quotient of division)
$\deg P > \deg Q + 1$	No OA (grows faster)

Other functions

Function	Asymptote
$1/x$	VA $x = 0$ , HA $y = 0$
$e^x$	HA $y = 0$ ( $x \to -\infty$ )
$\ln x$	VA $x = 0$
$\arctan x$	HA $y = \pm\pi/2$
$\tan x$	VA at $\pi/2 + k\pi$
$1/\sin x$	VA at $k\pi$

General procedure

VA: find points where $f$ is undefined (denom. = 0). Compute one-sided limit.
HA: compute $\lim_{x \to \pm\infty} f$ .
OA: if no HA, compute $m, b$ via the formulas.

Exercise list

36 exercises · 9 with worked solution (25%)

Application 24Understanding 2Modeling 8Challenge 1Proof 1

Ex. 47.1Application
Asymptotes of $f(x) = 1/(x - 2)$ .
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Ex. 47.2Application
Asymptotes of $f(x) = (3x + 1)/(x - 2)$ . (Ans: VA $x=2$ , HA $y=3$ .)
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Ex. 47.3ApplicationAnswer key
$f(x) = x^2/(x^2 - 1)$ — asymptotes?
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Ex. 47.4Application
$f(x) = (x^2 + 1)/x$ — asymptotes. (Ans: VA $x=0$ , OA $y=x$ .)
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Ex. 47.5ApplicationAnswer key
$f(x) = x + 1/x$ — asymptotes.
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Ex. 47.6Application
$f(x) = e^{-x}$ — HA at $x \to \infty$ ? (Ans: $y = 0$ .)
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Ex. 47.7Application
$f(x) = \tan x$ — vertical asymptotes. (Ans: $x = \pi/2 + k\pi$ .)
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Ex. 47.8Application
$f(x) = \arctan x$ — HA at $\pm\infty$ ? (Ans: $y = \pm\pi/2$ .)
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Ex. 47.9Application
$f(x) = \sqrt{x^2 + 1}$ — oblique asymptote? (Ans: $y = \pm x$ .)
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Ex. 47.10Application
$f(x) = (x^3 - 1)/(x^2 - 1)$ — asymptotes. (Ans: VA $x=-1$ , OA $y=x$ .)
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Ex. 47.11ApplicationAnswer key
$f(x) = (2x^2 - 3)/(x^2 + 1)$ — asymptotes.
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Ex. 47.12Application
$f(x) = (x^2 - 4)/(x - 2)$ — asymptotes? (Caution: removable at $x=2$ .)
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Ex. 47.13Application
Sketch $f(x) = (x+1)/(x-1)$ identifying VA and HA.
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Ex. 47.14Application
$f(x) = \ln x$ — VA at $x = 0$ ? Type? (Ans: $\lim \ln x = -\infty$ .)
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Ex. 47.15Application
$f(x) = e^x/x$ — asymptotes.
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Ex. 47.16ApplicationAnswer key
$f(x) = (x^2 + x)/(x + 1)$ — simplify and find asymptotes.
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Ex. 47.17Application
$f(x) = 1/\sin x$ — vertical asymptotes. (Ans: $x = k\pi$ .)
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Ex. 47.18Application
$f(x) = \tanh x$ — HA at $\pm\infty$ ? (Ans: $y = \pm 1$ .)
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Ex. 47.19Application
$f(x) = x \cdot e^{-x}$ — HA as $x \to +\infty$ .
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Ex. 47.20Application
$f(x) = (x^3 + 1)/(x^2 + 1)$ — oblique asymptote? (Ans: $y = x$ .)
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Ex. 47.21Application
Sketch $f(x) = 1/(x^2 - 4)$ identifying asymptotes.
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Ex. 47.22ApplicationAnswer key
Sketch $f(x) = x/(x^2 + 1)$ .
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Ex. 47.23Application
Sketch $f(x) = (x^2 + 2x)/(x - 1)$ with VA and OA.
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Ex. 47.24Application
Sketch $f(x) = 1/(x \ln x)$ — domain, asymptotes.
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Ex. 47.25ModelingAnswer key
In pharmacokinetics $C(t) \to 0$ — HA $y = 0$ . Verify for $C(t) = C_0 e^{-kt}$ .
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Ex. 47.26Modeling
In economics, average cost $\bar C(q) = C(q)/q$ has OA $y = c$ if $C(q) = c q + F$ (limit marginal cost).
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Ex. 47.27Modeling
Normal distribution $f(x) = e^{-x^2/2}/\sqrt{2\pi}$ — HA at $\pm\infty$ ?
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Ex. 47.28Modeling
In RC, $V(t) \to V_\infty$ — HA at $V = V_\infty$ .
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Ex. 47.29Modeling
Bode plot of $H(s) = K/(s+1)$ . Identify OAs in log-log for $\omega \to 0$ and $\omega \to \infty$ .
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Ex. 47.30ModelingAnswer key
Black-Scholes price $C(S, T) \to S - K e^{-rT}$ as $S \to \infty$ . OA in $S$ .
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Ex. 47.31Modeling
Logistic population growth $P(t) \to K$ — HA at carrying capacity.
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Ex. 47.32ModelingAnswer key
Hyperbola $x^2/a^2 - y^2/b^2 = 1$ — asymptotes $y = \pm (b/a)x$ . Verify.
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Ex. 47.33UnderstandingAnswer key
Show that $f$ may have at least 2 VAs (example: $\tan x$ has infinitely many).
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Ex. 47.34Understanding
Can a function have HA and OA at the same time (on the same side)? (Ans: No.)
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Ex. 47.35Challenge
$f(x) = (x^4 + 1)/(x^3 - 1)$ — asymptotes?
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Ex. 47.36Proof
Prove: if $f$ has OA $y = mx + b$ , then $\lim f/x = m$ and $\lim (f - mx) = b$ .

Sources

Calculus (Volume 1) — OpenStax · 2016 · §4.6.
Active Calculus — Boelkins · 2024 · §1.7.
APEX Calculus — Hartman · 2024 · §1.6.