j-james/math

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All my math notes, now in Markdown.

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More on Limits and using calculator for Numerical Derivatives and Integrals

Learning Targets

You should be able to

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$t(f(x), x, a, b)$

Note: Technically the theorem is based on real numbers, we do include infinity as a possible answer to give more information about answer.

Theorem

If $r > 0$ is a rational number, then \(\lim_{x \to \pm \infty} \frac{1}{x^r} = 0\)

Suppose we want to answer a limit question, and when we “plug in” the limit value, this is what we get. So, what would they equal?

\[\begin{matrix} \frac{0}{0}&\frac{\infty}{\infty}&0^{infty}\\ \frac{some\ number}{\infty}&\infty - \infty&0^0\\ \frac{\infty}{some\ number}&(0)(\infty)&1^{infty}\\ \frac{some\ number}{0}\\ \end{matrix}\] \[\begin{matrix} indeterminate&indeterminate&???\\ 0&indeterminate&indeterminate\\ \pm\infty&indeterminate&indeterminate\\ indeterminate\\ \end{matrix}\]

undefined: we don’t know
indeterminute: we can find out