Sunghee Yun (He/Him)

Co-Founder & CTO @ Erudio Bio, Inc., Leader of Silicon Valley Privacy Preserving AI Forum (K-PAI), Advisor @ Korean American Semiconductor Professional Alliance (KASPA), CGO / Global Managing Partner @ LULUMEDIC, KFAS-Salzburg Global Leadership Initiative Fellow @ Salzburg Global Seminar, AI-Korean Medicine Integration Initiative Task Force Member @ The Association of Korean Medicine, Director of AI Semiconductor @ K-BioX, Visiting Professor @ Sogang University, Advisory Professor @ DGIST, Global Advisory Board Member @ Innovative Future Brain-Inspired Intelligence System Semiconductor of Sogang University, Network Expert @ Gerson Lehrman Group, and Chief Business Development Officer @ WeStory.ai

4 minute read

Short daily sessions — 15–20 min, casual, no pressure. 🧠
Think of it like a daily stretch before a game. The goal is exposure, not perfection.
The more times you bump into a concept, the more natural it becomes.
“The best way to understand a limit is to bump into it over and over —
not to stare at it once for three hours.” — Daddy

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March 2026

March 9, 2026

📅 Mon, March 9, 2026  |  Topics – Limits at 0 and $\infty$ · Series convergence · FTC Part 1

Part 1 — Limits

Find the value of each limit (write $\infty$, $-\infty$, or DNE when appropriate).

  1. $\displaystyle\lim_{x\to\infty} \dfrac{x^2 - 3x}{-3x^2 + 4x + 1}$
  2. $\displaystyle\lim_{x\to\infty} \dfrac{x^2 - 3x}{-3x^2 + 4x}$
  3. $\displaystyle\lim_{x\to 0} \dfrac{x^2 - 3x}{-3x^2 + 4x}$
  4. $\displaystyle\lim_{x\to 0^+} x\ln x$
  5. $\displaystyle\lim_{x\to 0^+} x^2 \ln x$
  6. $\displaystyle\lim_{x\to 0^+} x^{10} \ln x$
  7. $\displaystyle\lim_{x\to 0^+} x^{0.01} \ln x$
  8. $\displaystyle\lim_{x\to 0^+} x^{-1} \ln x$    (careful — which dominates?)
  9. $\displaystyle\lim_{x\to 0^+} x^{-0.01} \ln x$    (subtle!)
  10. $\displaystyle\lim_{x\to\infty} x\ln x$
  11. $\displaystyle\lim_{x\to\infty} x^2 \ln x$
  12. $\displaystyle\lim_{x\to\infty} x^{10} \ln x$
  13. $\displaystyle\lim_{x\to\infty} x^{0.01} \ln x$
  14. $\displaystyle\lim_{x\to\infty} x^{-1} \ln x$
  15. $\displaystyle\lim_{x\to\infty} x^{-0.01} \ln x$    (compare growth rates carefully)
  16. $\displaystyle\lim_{x\to\infty} \dfrac{\ln(100 + e^{3x}) + 10^{10^{10}}}{5x}$
  17. $\displaystyle\lim_{x\to\infty} \dfrac{\ln(100 + e^{3x}) - \ln 101}{5x}$
  18. $\displaystyle\lim_{x\to 0} \dfrac{\ln(100 + e^{3x}) - \ln 101}{5x}$
  19. $\displaystyle\lim_{x\to 0} \dfrac{\ln(100 + e^{3x})}{5x}$    (does this even exist?)
  20. $\displaystyle\lim_{n\to\infty} (-1)^n$
▶ Answers — Part 1
  1. $-\dfrac{1}{3}$ — divide top and bottom by $x^2$; only leading coefficients survive
  2. $-\dfrac{1}{3}$ — same; the $+1$ in the denominator is irrelevant at $\infty$
  3. $-\dfrac{3}{4}$ — factor out $x$: $\frac{x(x-3)}{x(-3x+4)}$, cancel $x$, plug in $x=0$: $\frac{0-3}{0+4} = -\frac{3}{4}$
  4. $0$ — rewrite as $\frac{\ln x}{x^{-1}}$; L'Hôpital: $\frac{1/x}{-x^{-2}} = -x \to 0$
  5. $0$ — $x^2$ kills $\ln x$ even faster; for any $p>0$: $\lim_{x\to 0^+} x^p\ln x = 0$
  6. $0$ — same rule, $p=10$
  7. $0$ — same rule, $p=0.01>0$; polynomial always beats log
  8. $-\infty$ — $\frac{\ln x}{x} \xrightarrow{x\to 0^+} \frac{-\infty}{0^+} = -\infty$ (denominator $\to 0^+$ makes it blow up)
  9. $-\infty$ — $p=-0.01<0$ so $x^{-0.01}\to +\infty$ while $\ln x\to -\infty$; product $\to -\infty$
  10. $\infty$ — both $x\to\infty$ and $\ln x\to\infty$; no competition
  11. $\infty$
  12. $\infty$
  13. $\infty$ — even a tiny positive power $\times \ln x \to \infty$
  14. $0$ — $\frac{\ln x}{x}\to 0$; polynomial growth crushes logarithm at $\infty$
  15. $0$ — $\frac{\ln x}{x^{0.01}}\to 0$; any polynomial power beats $\ln x$ at $\infty$
  16. $\dfrac{3}{5}$ — the constant $10^{10^{10}}$ is irrelevant at $\infty$; $\ln(100+e^{3x})\approx 3x$ for large $x$, so limit $=\frac{3x}{5x}=\frac{3}{5}$
  17. $\dfrac{3}{5}$ — $-\ln 101$ is a fixed constant, irrelevant at $\infty$; same reasoning
  18. $\dfrac{3}{505}$ — at $x=0$: $\frac{0}{0}$ form; L'Hôpital: $\dfrac{3e^{3x}/(100+e^{3x})}{5}\Big|_{x=0} = \dfrac{3/101}{5} = \dfrac{3}{505}$
  19. DNE — numerator $\to\ln 101\neq 0$ while denominator $\to 0$; from the right $\to+\infty$, from the left $\to-\infty$
  20. DNE — $(-1)^n$ oscillates between $+1$ and $-1$ forever; no single value

Part 2 — Series

– Converge or Diverge?

For each series – does it converge or diverge? If it converges, give the exact value if you can. If it diverges, write $\infty$, $-\infty$, or “oscillates.”

  1. $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n}$
  2. $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2}$
  3. $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^4}$
  4. $\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n}$
  5. $\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^{n}}{2n+1}$
  6. $\displaystyle\sum_{n=1}^{\infty} (-1)^n$
  7. $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n!}$
  8. $\displaystyle\sum_{n=1}^{\infty} 2^{n-1}$
  9. $\displaystyle\sum_{n=1}^{\infty} (0.5)^{n-1}$
  10. $\displaystyle\sum_{n=1}^{\infty} (0.25)^{n-1}$
  11. $\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{1}{3}\right)^{n-1}$
  12. $\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{e^n}$
  13. $\displaystyle\sum_{n=1}^{\infty} \dfrac{e^n}{n!}$
  14. $\displaystyle\sum_{n=1}^{\infty} \dfrac{n^2}{e^n}$
  15. $\displaystyle\sum_{n=1}^{\infty} \dfrac{n^{1000}}{e^n}$
▶ Answers — Part 2
  1. Diverges ($\infty$) — the harmonic series; famously diverges (grows like $\ln n$)
  2. Converges $= \dfrac{\pi^2}{6} \approx 1.6449$ — Basel problem (Euler, 1734); $p$-series with $p=2>1$
  3. Converges $= \dfrac{\pi^4}{90} \approx 1.0823$ — $p$-series with $p=4>1$
  4. Converges $= \ln 2 \approx 0.6931$ — alternating harmonic series (Leibniz test)
  5. Converges $= \dfrac{\pi}{4} \approx 0.7854$ — Gregory–Leibniz formula for $\pi$! (from $\arctan 1 = \pi/4$)
  6. Diverges (oscillates) — partial sums alternate $-1, 0, -1, 0, \ldots$; no limit
  7. Converges $= e-1 \approx 1.7183$ — Taylor series for $e^x$ at $x=1$, minus the $n=0$ term
  8. Diverges ($\infty$) — geometric with ratio $r=2>1$
  9. Converges $= 2$ — geometric: $\dfrac{1}{1-0.5} = 2$
  10. Converges $= \dfrac{4}{3}$ — geometric: $\dfrac{1}{1-0.25} = \dfrac{4}{3}$
  11. Converges $= \dfrac{3}{2}$ — geometric: $\dfrac{1}{1-\frac{1}{3}} = \dfrac{3}{2}$
  12. Diverges ($\infty$) — ratio test: $\dfrac{(n+1)!}{e^{n+1}}\cdot\dfrac{e^n}{n!} = \dfrac{n+1}{e}\to\infty$
  13. Converges $= e^e - 1 \approx 14.77$ — ratio test: $\dfrac{e}{n+1}\to 0<1$; relates to Taylor expansion of $e^e$
  14. Converges — ratio test: $\dfrac{(n+1)^2}{n^2 \cdot e}\to\dfrac{1}{e}<1$
  15. Converges — ratio test: limit $\to\dfrac{1}{e}<1$; exponential always beats any polynomial power

Part 3 — FTC, Part 1

Find each derivative. Hint – you do NOT need to evaluate the integral!

  1. $\displaystyle\frac{d}{dx} \int_{0}^{x} \sqrt{\sin^4 t + 234}\; dt$
  2. $\displaystyle\frac{d}{dx} \int_{-1000}^{x} \sqrt{\sin^4 t + 234}\; dt$
  3. $\displaystyle\frac{d}{dx} \int_{10^{10}}^{x} \sqrt{\sin^4 t + 234}\; dt$

👀 Look carefully at all three answers. What do you notice about the lower limit?

▶ Answers — Part 3
All three give the same answer: $$\frac{d}{dx}\int_a^x \sqrt{\sin^4 t + 234}\;dt \;=\; \sqrt{\sin^4 x + 234}$$

The lower limit $a$ — whether $0$, $-1000$, or $10^{10}$ — only contributes a fixed constant to the antiderivative, and constants vanish under differentiation. FTC Part 1: $\;\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt = f(x)\;$ for any constant $a$.

Also: inside the integral the dummy variable is $t$, not $x$. Writing $\sqrt{\sin^4 x+234}$ inside the integral while differentiating with respect to $x$ would be a sign confusion — that's exactly why the dummy variable matters!

New sessions are prepended at the top of each month’s section — most recent first.

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