Calc I: Practice for quiz 1

Here are a few more practice problems for our first quiz this Friday.

Problems

  1. Use a little algebra together with the limit laws to show that \[ \lim_{x\to2}\frac{x-2}{x^{3} - x^{2} - x - 2} = \frac{1}{7}. \] Be sure to write your solution out fully using correct mathematical notation and connectives.
  1. Let \(f(x) = 1-x-x^3\). Write a complete sentence referring to the intermediate value theorem proving that \(f\) has a root between \(x=0\) and \(x=1\).

  1. Find the values of the following limits quickly and efficiently. You need write only the answer and you generally don’t need to simplify.

    1. \(\displaystyle \lim_{x\to3} \frac{x^5 - x^2 - x}{2x^2 - 5}.\)
    2. \(\displaystyle \lim_{x\to\infty} \frac{x^2 - x}{2x^2 - 5}.\)
    3. \(\displaystyle \lim_{x\to3}\frac{x^2-6}{(x-3)^2}\)

  1. The graph of a function \(f\) is shown in Figure 1 below. Use that graph to evaluate the following:

    1. \(\lim_{x\to 0^-}f(x)\)
    2. \(\lim_{x\to 0^+}f(x)\)
    3. \(\lim_{x\to 0}f(x)\)
    4. \(f(0)\)
    5. \(\lim_{x\to 4^-}f(x)\)
    6. \(\lim_{x\to 4^+}f(x)\)
    7. \(\lim_{x\to 4}f(x)\)
    8. \(f(4)\)
    9. \(\lim_{x\to\infty}f(x)\)
Figure 1: The graph for problem 4