Calculus 1 — Limits & Derivatives

Limits — sneaking up on a value

sin(x)/x at x = 0 is 0/0 — meaningless. But near 0 it is perfectly well-behaved. Make x small and watch it commit to 1:

r_{1} = \frac{\mathrm{sin}\left(0.1\right)}{0.1}

0.9983

getting close…

r_{2} = \frac{\mathrm{sin}\left(0.001\right)}{0.001}

1.0000

closer…

r_{3} = \frac{\mathrm{sin}\left(10^{-8}\right)}{10^{-8}}

1

the limit is 1

The most famous limit in mathematics: compound interest, compounded continuously. (1 + 1/n)ⁿ as n grows doesn't blow up — it converges to e:

e_{approx} = \left(1 + \frac{1}{1000000}\right)^{1000000}

2.7183

e ≈ 2.71828… — try larger n