# j-james/math All my math notes, now in Markdown.

View the Project on GitHub j-james/math

# More on Limits and using calculator for Numerical Derivatives and Integrals

## Learning Targets

You should be able to

• Evaluate limits algebraically and numerically, with and without calculator
• Numerically calculate derivatives and integrals using nDeriv and fnInt on calculator

## Concepts / Definitions

### Theorem for a Limit

$\lim_{x \to a} f(x) = L \quad iff \quad \lim_{x \to a^-} f(x) = L = \lim_{x \to a^+} f(x)$

### Theorem

$\lim_{x \to \infty} \frac{\sin{x}}{x} = 1 \qquad \lim_{x \to 0} \frac{\cos{x}-1}{x} = 0$

### Derivative and Integral buttons

From Graphing Screen:
Graph $f(x)$, $2^{nd}$ TRACE (calc), 6 deriv, 7 integral
From Main Screen:
MATH, 8nDeriv$(f(x), x, x_a)$, 9fnIn$t(f(x), x, a, b)$

If $r > 0$ is a rational number, then $$\lim_{x \to \pm \infty} \frac{1}{x^r} = 0$$
$\begin{matrix} \frac{0}{0}&\frac{\infty}{\infty}&0^{infty}\\ \frac{some\ number}{\infty}&\infty - \infty&0^0\\ \frac{\infty}{some\ number}&(0)(\infty)&1^{infty}\\ \frac{some\ number}{0}\\ \end{matrix}$ $\begin{matrix} indeterminate&indeterminate&???\\ 0&indeterminate&indeterminate\\ \pm\infty&indeterminate&indeterminate\\ indeterminate\\ \end{matrix}$