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Lesson 44 — One-Sided and Infinite Limits

Right- and left-hand limits. Infinite limits and limits at infinity. Existence via one-sided limits.

Used in: 2.º ano do EM (16-17 anos) · Equiv. Math II japonês §limites unilaterais · Equiv. Analysis-Vorkurs alemão

lim⁡x→af(x)=L  ⟺  lim⁡x→a+f(x)=lim⁡x→a−f(x)=L\lim_{x \to a} f(x) = L \iff \lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x) = L
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Rigorous notation, full derivation, hypotheses

Rigorous definitions

One-sided limits

Existence theorem via one-sided limits

lim⁡x→af(x)=L  ⟺  lim⁡x→a+f(x)=lim⁡x→a−f(x)=L\lim_{x \to a} f(x) = L \iff \lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x) = L.

Infinite limits

lim⁡x→af(x)=+∞\lim_{x \to a} f(x) = +\infty: ∀M>0,∃δ>0:0<∣x−a∣<δ⇒f(x)>M\forall M > 0, \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow f(x) > M.

Analogous for −∞-\infty and one-sided.

Limits at infinity

lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L: ∀\eps>0,∃N:x>N⇒∣f(x)−L∣<\eps\forall \eps > 0, \exists N : x > N \Rightarrow |f(x) - L| < \eps.

lim⁡x→∞f(x)=+∞\lim_{x \to \infty} f(x) = +\infty: ∀M,∃N:x>N⇒f(x)>M\forall M, \exists N : x > N \Rightarrow f(x) > M.

Quantifier table

TypeForm
lim⁡x→a=L\lim_{x \to a} = L∀\eps,∃δ\forall \eps, \exists \delta
lim⁡x→a+=L\lim_{x \to a^+} = L∀\eps,∃δ\forall \eps, \exists \delta, x>ax > a
lim⁡x→a=∞\lim_{x \to a} = \infty∀M,∃δ\forall M, \exists \delta
lim⁡x→∞=L\lim_{x \to \infty} = L∀\eps,∃N\forall \eps, \exists N
lim⁡x→∞=∞\lim_{x \to \infty} = \infty∀M,∃N\forall M, \exists N

"Pass-through" rules for P/QP/Q as x→∞x \to \infty

deg⁡P\deg P vs deg⁡Q\deg Qlim⁡\lim
deg⁡P>deg⁡Q\deg P > \deg Q±∞\pm\infty
deg⁡P=deg⁡Q\deg P = \deg Qratio of leading coeffs
deg⁡P<deg⁡Q\deg P < \deg Q00

Exercise list

40 exercises · 10 with worked solution (25%)

Application 28Understanding 3Modeling 6Proof 3
  1. Ex. 44.1ApplicationAnswer key
    lim⁡x→0+1/x\lim_{x \to 0^+} 1/x. (Ans: +∞+\infty.)
    Solve online
  2. Ex. 44.2ApplicationAnswer key
    lim⁡x→0−1/x\lim_{x \to 0^-} 1/x.
    Solve online
  3. Ex. 44.3ApplicationAnswer key
    lim⁡x→2+1/(x−2)\lim_{x \to 2^+} 1/(x - 2).
    Solve online
  4. Ex. 44.4Application
    lim⁡x→2−1/(x−2)\lim_{x \to 2^-} 1/(x - 2).
    Solve online
  5. Ex. 44.5ApplicationAnswer key
    lim⁡x→∞1/x\lim_{x \to \infty} 1/x.
    Solve online
  6. Ex. 44.6Application
    lim⁡x→∞(3x+1)/(x+5)\lim_{x \to \infty} (3x + 1)/(x + 5). (Ans: 3.)
    Solve online
  7. Ex. 44.7Application
    lim⁡x→∞(x2+1)/(x+1)\lim_{x \to \infty} (x^2 + 1)/(x + 1).
    Solve online
  8. Ex. 44.8Application
    lim⁡x→∞e−x\lim_{x \to \infty} e^{-x}.
    Solve online
  9. Ex. 44.9ApplicationAnswer key
    lim⁡x→∞(ln⁡x)/x\lim_{x \to \infty} (\ln x)/x.
    Solve online
  10. Ex. 44.10ApplicationAnswer key
    lim⁡x→0+ln⁡x\lim_{x \to 0^+} \ln x. (Ans: −∞-\infty.)
    Solve online
  11. Ex. 44.11Application
    lim⁡x→(π/2)−tan⁡x\lim_{x \to (\pi/2)^-} \tan x.
    Solve online
  12. Ex. 44.12ApplicationAnswer key
    lim⁡x→(π/2)+tan⁡x\lim_{x \to (\pi/2)^+} \tan x.
    Solve online
  13. Ex. 44.13Application
    lim⁡x→0∣x∣/x\lim_{x \to 0} |x|/x — one-sided and bilateral.
    Solve online
  14. Ex. 44.14Application
    lim⁡x→1⌊x⌋\lim_{x \to 1} \lfloor x \rfloor — one-sided.
    Solve online
  15. Ex. 44.15Application
    lim⁡x→∞x2+3−x\lim_{x \to \infty} \sqrt{x^2 + 3} - x.
    Solve online
  16. Ex. 44.16Application
    lim⁡x→−∞x2+3+x\lim_{x \to -\infty} \sqrt{x^2 + 3} + x. (Ans: 0.)
    Solve online
  17. Ex. 44.17Application
    lim⁡x→∞(x4−3x2)/(2x4+1)\lim_{x \to \infty} (x^4 - 3x^2)/(2x^4 + 1). (Ans: 1/21/2.)
    Solve online
  18. Ex. 44.18Application
    lim⁡x→∞arctan⁡x\lim_{x \to \infty} \arctan x. (Ans: π/2\pi/2.)
    Solve online
  19. Ex. 44.19Application
    lim⁡x→−∞arctan⁡x\lim_{x \to -\infty} \arctan x.
    Solve online
  20. Ex. 44.20ApplicationAnswer key
    lim⁡x→0+xln⁡x\lim_{x \to 0^+} x \ln x. (Ans: 0.)
    Solve online
  21. Ex. 44.21Application
    f(x)={x+1x<0x2+1x≥0f(x) = \begin{cases} x + 1 & x < 0 \\ x^2 + 1 & x \geq 0 \end{cases}. Does lim⁡x→0\lim_{x \to 0} exist?
    Solve online
  22. Ex. 44.22Application
    f(x)={2xx≤1x+1x>1f(x) = \begin{cases} 2x & x \leq 1 \\ x + 1 & x > 1 \end{cases}. lim⁡x→1\lim_{x \to 1}? (Ans: 2.)
    Solve online
  23. Ex. 44.23ApplicationAnswer key
    f(x)={sin⁡x/xx≠00x=0f(x) = \begin{cases} \sin x / x & x \neq 0 \\ 0 & x = 0 \end{cases}. lim⁡x→0=?\lim_{x \to 0} = ?
    Solve online
  24. Ex. 44.24Application
    Find aa for f(x)={axx<1x2x≥1f(x) = \begin{cases} ax & x < 1 \\ x^2 & x \geq 1 \end{cases} to have a limit at x=1x = 1.
    Solve online
  25. Ex. 44.25Application
    Does lim⁡x→0sin⁡(1/x)\lim_{x \to 0} \sin(1/x) exist? Justify.
    Solve online
  26. Ex. 44.26Application
    f(x)=e1/xf(x) = e^{1/x}. Compute lim⁡x→0+\lim_{x \to 0^+} and lim⁡x→0−\lim_{x \to 0^-}.
    Solve online
  27. Ex. 44.27Application
    lim⁡x→0+e−1/x\lim_{x \to 0^+} e^{-1/x} and lim⁡x→0−e−1/x\lim_{x \to 0^-} e^{-1/x}.
    Solve online
  28. Ex. 44.28Application
    ff has lim⁡x→a+=L1\lim_{x \to a^+} = L_1 and lim⁡x→a−=L2\lim_{x \to a^-} = L_2 with L1≠L2L_1 \neq L_2. Does the bilateral lim⁡\lim exist? (Ans: No.)
    Solve online
  29. Ex. 44.29Modeling
    In vibration mechanics, damped oscillations: A(t)=A0e−γtcos⁡(ωt)A(t) = A_0 e^{-\gamma t} \cos(\omega t). lim⁡t→∞A(t)=?\lim_{t \to \infty} A(t) = ?
    Solve online
  30. Ex. 44.30Modeling
    In pharmacokinetics, C(t)=C0e−ktC(t) = C_0 e^{-kt}. lim⁡t→∞C(t)=0\lim_{t \to \infty} C(t) = 0 — interpret.
    Solve online
  31. Ex. 44.31Modeling
    In a capacitor: V(t)=V∞(1−e−t/RC)V(t) = V_\infty (1 - e^{-t/RC}). lim⁡t→∞V(t)=V∞\lim_{t \to \infty} V(t) = V_\infty.
    Solve online
  32. Ex. 44.32Modeling
    In economics, C(q)/q→cC(q)/q \to c (average cost → marginal cost). Model lim⁡q→∞\lim_{q \to \infty}.
    Solve online
  33. Ex. 44.33Modeling
    System response with transfer function H(s)=K/(s+1)H(s) = K/(s+1). DC gain =lim⁡s→0H(s)=K= \lim_{s \to 0} H(s) = K.
    Solve online
  34. Ex. 44.34Modeling
    Boltzmann distribution: p(E)∝e−E/kTp(E) \propto e^{-E/kT}. lim⁡E→∞p=0\lim_{E \to \infty} p = 0, lim⁡T→0+\lim_{T \to 0^+} concentrates on Emin⁡E_{\min}.
    Solve online
  35. Ex. 44.35Understanding
    Show that if lim⁡x→a+=lim⁡x→a−=L\lim_{x \to a^+} = \lim_{x \to a^-} = L, then lim⁡x→a=L\lim_{x \to a} = L.
    Solve online
  36. Ex. 44.36Understanding
    Construct a function ff with lim⁡x→0+=1\lim_{x \to 0^+} = 1 and lim⁡x→0−=−1\lim_{x \to 0^-} = -1.
    Solve online
  37. Ex. 44.37Understanding
    Show that lim⁡x→∞P(x)/Q(x)\lim_{x \to \infty} P(x)/Q(x) depends on the leading degrees (3 cases).
    Solve online
  38. Ex. 44.38ProofAnswer key
    Prove lim⁡x→∞P(x)/Q(x)=0\lim_{x \to \infty} P(x)/Q(x) = 0 if deg⁡P<deg⁡Q\deg P < \deg Q.
  39. Ex. 44.39Proof
    Prove lim⁡x→0+1/x=+∞\lim_{x \to 0^+} 1/x = +\infty via ε-M.
  40. Ex. 44.40Proof
    Prove lim⁡x→∞arctan⁡x=π/2\lim_{x \to \infty} \arctan x = \pi/2.

Sources

Updated on 2026-04-30 · Author(s): Clube da Matemática

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