# Limit definition w/ graphs

Today's activity introduces mathematical "zooming".

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

1. 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 $$x = 2$$.
2. 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?
3. 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?
4. How can we find the "largest" domains above without guessing-and-checking?

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

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 $$x=1$$.
2. 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?
3. 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?
4. 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.