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: \( 0 \lt x \lt \dfrac{4}{3} \)
centered at \(x=1\):
\(
\begin{align*}
\frac{2}{3} & \lt x \lt \frac{4}{3} \\
1 - \frac{1}{3} & \lt x \lt 1 + \frac{1}{3}
\end{align*}
\)
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: \( \dfrac{18}{19} \lt x \lt \dfrac{22}{21} \)
centered at \(x=1\):
\(
\begin{align*}
\frac{20}{21} & \lt x \lt \frac{22}{21} \\
1-\frac{1}{21} & \lt x \lt 1+\frac{1}{21}
\end{align*}
\)
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: \( \dfrac{198}{199} \lt x \lt \dfrac{202}{201} \)
centered at \(x=1\):
\(
\begin{align*}
\frac{200}{201} & \lt x \lt \frac{202}{201} \\
1-\frac{1}{201} & \lt x \lt 1+\frac{1}{201}
\end{align*}
\)
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)