Important limits — lim(sin x - x) = 1, lim((1+1 - n)ⁿ) = e

Part 1 — $\lim_{x\to 0}\dfrac{\sin x}{x}=1$

HOW to derive it — the squeeze (sandwich) argument

Look at a unit circle, with a small angle $x>0$ (radians) at the centre.

Compare three areas for $0<x<\frac\pi2$:

1. Triangle inside (vertices: centre, $(1,0)$, point on circle): area $=\frac12\cdot 1\cdot \sin x=\frac{\sin x}{2}$.
   Why this step? Triangle area $=\frac12 \times \text{base} \times \text{height}$; base $=1$ (radius), height $=\sin x$.
2. Circular sector: area $=\frac12 r^2 x = \frac{x}{2}$.
   Why this step? Sector area $=\frac12 r^2\theta$ with $r=1$, $\theta=x$.
3. Triangle outside (using the tangent): area $=\frac12\cdot1\cdot\tan x=\frac{\tan x}{2}$.
   Why this step? The tangent line gives a bigger right triangle of height $\tan x$.

Geometrically: inner triangle $\subset$ sector $\subset$ outer triangle, so $\frac{\sin x}{2}\le \frac{x}{2}\le \frac{\tan x}{2}.$

Multiply by $2$ and divide by $\sin x>0$: $1\le \frac{x}{\sin x}\le \frac{1}{\cos x}.$ Why this step? Dividing by the positive quantity $\sin x$ keeps inequalities the same direction; $\tan x/\sin x = 1/\cos x$.

Take reciprocals (flips inequalities): $\cos x \le \frac{\sin x}{x}\le 1.$

Now let $x\to 0$: $\cos x\to 1$ and the right side is $1$. By the Squeeze Theorem the middle is trapped: $\boxed{\lim_{x\to 0}\frac{\sin x}{x}=1.}$

Part 2 — $\lim_{n\to\infty}\left(1+\dfrac1n\right)^n=e$

HOW to derive — turn the exponent into a log

The clean way: take the natural log and use a Taylor/known limit. $\ln\!\left(1+\tfrac1n\right)^n = n\ln\!\left(1+\tfrac1n\right).$ Let $h=\frac1n\to0$. Then $n\ln(1+h)=\frac{\ln(1+h)}{h}.$ Why this step? $n=1/h$, so multiplying by $n$ is dividing by $h$ — now it's a $0/0$ limit we can evaluate.

Use the standard limit $\lim_{h\to0}\frac{\ln(1+h)}{h}=1$ (this is the derivative of $\ln$ at $1$, since $\frac{\ln(1+h)-\ln1}{h}\to \frac{d}{dx}\ln x\big|_{x=1}=1$).

So $\ln(\text{expression})\to 1$, therefore the expression $\to e^1=e$. $\blacksquare$

Common mistakes (steel-manned)

Recall

Connections

• Squeeze (Sandwich) Theorem — the engine behind $\frac{\sin x}{x}$.
• Radian Measure — why the "1" needs radians.
• Derivative of sin and cos — built directly from this limit.
• The Number e and ln — defined by the second limit.
• Indeterminate Forms — $0/0$ and $1^\infty$.
• L'Hôpital's Rule — an alternative (circular for these, so we used geometry/log).
• Taylor Series — $\sin x=x-\frac{x^3}{6}+\dots$, $e^x=\sum x^k/k!$.

Flashcards

Concept Map

Hinglish (regional understanding)