Further Calculus

$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}.$

Always check the indeterminate form FIRST — applying L'Hôpital to a non-indeterminate form gives a wrong answer.

Worked example. $\lim_{x \to 0} \frac{\sin x}{x}$.

Form: $0/0$ ✓. Apply: $\lim \frac{\cos x}{1} = 1$.

Worked example (iterated). $\lim_{x \to 0} \frac{1 - \cos x}{x^2}$.

Form $0/0$. Apply: $\lim \frac{\sin x}{2x}$ still $0/0$. Apply again: $\lim \frac{\cos x}{2} = 1/2$.

Other forms.

• $0 \cdot \infty$: rewrite as $0/0$ or $\infty/\infty$.
• $\infty - \infty$: combine into a single fraction.
• $1^\infty, 0^0, \infty^0$: take $\ln$, evaluate, then exponentiate.

Worked example. $\lim_{x \to 0^+} x \ln x$ (form $0 \cdot (-\infty)$).

Rewrite: $\lim \frac{\ln x}{1/x}$ ($-\infty/\infty$). Apply: $\lim \frac{1/x}{-1/x^2} = \lim (-x) = 0$.

• Verify indeterminate form BEFORE applying.
• May need iteration.
• Other forms reduce via algebra or log/exp.

Differentiability. $f'(a)$ exists ⇒ $f$ continuous at $a$. The converse is FALSE: $|x|$ is continuous at 0 but not differentiable.

Maclaurin series. Expansion about $x = 0$:

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n.$

Standard Maclaurin series (memorise):

$e^x = 1 + x + \tfrac{x^2}{2!} + \tfrac{x^3}{3!} + \cdots$ (all $x$).

$\sin x = x - \tfrac{x^3}{3!} + \tfrac{x^5}{5!} - \cdots$ (all $x$).

$\cos x = 1 - \tfrac{x^2}{2!} + \tfrac{x^4}{4!} - \cdots$ (all $x$).

$\ln(1 + x) = x - \tfrac{x^2}{2} + \tfrac{x^3}{3} - \cdots$ ($-1 < x \le 1$).

$(1 + x)^n = 1 + nx + \tfrac{n(n-1)}{2!}x^2 + \cdots$ ($|x| < 1$, general $n$).

Worked example. Derive the first four non-zero terms of $\arctan x$.

$f(x) = \arctan x$, $f(0) = 0$. $f'(x) = 1/(1 + x^2) = 1 - x^2 + x^4 - x^6 + \cdots$.

Integrate term-by-term: $\arctan x = x - x^3/3 + x^5/5 - x^7/7 + \cdots$.

Worked example (composition). Find Maclaurin series of $e^{-x^2}$ up to $x^4$.

$e^u = 1 + u + u^2/2 + \cdots$ with $u = -x^2$. So $e^{-x^2} = 1 - x^2 + x^4/2 - \cdots$.

• Continuous ⇏ differentiable.
• Maclaurin: derivatives at 0 build series.
• Compose, integrate or differentiate known series to shortcut.