Today's activity introduces mathematical "zooming".

1. Consider \(f(x) = \dfrac{2 x^2-x-6}{x-2}, x\neq2\) and notice \(\displaystyle\lim_{x \to 2} f(x) = 7\).

- If \(6 < f(x) < 8\), what domain makes \(f(x)\) go off the "sides" of the calculator window?

What is the largest window you can make? Hint: Stay close to \(x = 2\).

\( 1.5 \lt x \lt 2.5 \) - If \(6.9 < f(x) < 7.1\), what domain makes \(f(x)\) go off the "sides" of the calculator window? What is the largest window you can make?

\( 1.95 \lt x \lt 2.05 \) - If \(6.99 < f(x) < 7.01\), what domain makes \(f(x)\) go off the "sides" of the calculator window? What is the largest window you can make?

\( 1.995 \lt x \lt 2.005 \) - How can we find the "largest" domains above without guessing-and-checking?

Set \(f(x)\) equal to lower/upper bound and solve for \(x\).

2. Consider \(g(x) = \dfrac{x}{x-2}, x \neq 2\) and notice \(\displaystyle\lim_{x \to 1} g(x) = -1\).

- If \(-2 < g(x) < 0\), what domain makes \(g(x)\) go off the "sides" of the calculator window? What is the largest window you can make? Hint: Stay close to \(x=1\).

largest: \( 0 \lt x \lt \dfrac{4}{3} \)

symmetrical:

\( \begin{align*} \frac{2}{3} & \lt x \lt \frac{4}{3} \\ 1 - \frac{1}{3} & \lt x \lt 1 + \frac{1}{3} \end{align*} \) - If \(-1.1 < g(x) < -0.9\), what domain makes \(g(x)\) go off the "sides" of the calculator window? What is the largest window you can make?

largest: \( \dfrac{18}{19} \lt x \lt \dfrac{22}{21} \)

symmetrical:

\( \begin{align*} \frac{20}{21} & \lt x \lt \frac{22}{21} \\ 1-\frac{1}{21} & \lt x \lt 1+\frac{1}{21} \end{align*} \) - If \(-1.01 < g(x) < -0.99\), what domain makes \(g(x)\) go off the "sides" of the calculator window? What is the largest window you can make?

largest: \( \dfrac{198}{199} \lt x \lt \dfrac{202}{201} \)

symmetrical:

\( \begin{align*} \frac{200}{201} & \lt x \lt \frac{202}{201} \\ 1-\frac{1}{201} & \lt x \lt 1+\frac{1}{201} \end{align*} \) - For parts a-c, what are the largest windows that will work that are
**centered**at \(x=1\)? For example, \([0,3]\) is not centered at \(x=1\), but \([0,2]\) is.

(see above)