Limit definition w/ graphs

Today's activity introduces mathematical "zooming".

1. Suppose that \(f(x) = \dfrac{2 x^2-x-6}{x-2}, x\neq2\).

  1. If \(6 < f(x) < 8\), what domain makes \(f(x)\) go off the "sides" of the calculator window?
    What is the biggest 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 biggest 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 biggest window you can make?
  4. Without guess and checking, how could you find the "biggest" domains above?

2. Suppose that \(g(x) = \dfrac{x}{x-2}, x \neq 2\).

  1. If \(-2 < g(x) < 0\), what domain makes \(g(x)\) go off the "sides" of the calculator window? What is the biggest 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 biggest 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 biggest window you can make?
  4. For parts a-c, what are the biggest windows that will work that are symmetrical to \(x=1\)? For example, \([0,3]\) is not symmetrical to \(x=1\), but \([0,2]\) is.