Calc I - Review for Quiz 1

We have our first quiz this Friday, Jan 24. All the problems on that quiz will likely look like something you see on this problem sheet, though this sheet is a bit longer than the quiz will be.

Note that there is a link at the bottom of this sheet labeled either “Start Discussion” or “Continue Discussion”, depending on the status of those discussions. Following that link will take you to my forum where you can create an account with your UNCA email address. You can ask questions and/or post responses and answers there.

  1. Curious about the following limit, \[\lim _{x\to 0} (1+x)^{2/x},\] I used my computer to plug in several values of \(x\) that are close to \(0\) but not equal to \(0\). The results are shown in the table below.

    \(x\) 0.1 0.01 0.001 0.0001 0.00001
    \(f(x)\) 6.7275 7.31602 7.38168 7.38832 7.38898

    Based on those computations, can you make a conjecture as to the approximate value of the limit? Be sure to indicate how many digits you believe to be correct and why.

  2. The graph of \[f(x) = \frac{x - 1}{x^{3} - x^{2} + x - 1}\] is shown in Figure 1 below.

    1. Judging from the figure, what do you suppose is the value of \(\lim_{x\to1}f(x)\)?
    2. Use a little algebra together with the limit laws to prove that your guess is correct.

  3. The Complete graph of a function \(f\) is shown in Figure 2 below. At each of the points \(a=-1\), \(a=1\), \(a=2\) and \(a=4\), find the value of

  1. \(f(a)\),
  2. \(\displaystyle \lim\limits_{x\to a^-} f(x)\),
  3. \(\displaystyle \lim\limits_{x\to a^+} f(x)\), and
  4. \(\displaystyle \lim\limits_{x\to a} f(x)\).

  1. Continuing with Figure 2, state one clear reason why \(f\) is discontinuous at each of the points \(a=-1\), \(a=1\), \(a=2\) and \(a=4\).

  2. Compute each of the following limits. For part (a) make sure to write your solution out carefully. I’m primarily interested in answers for the others.

  1. \(\displaystyle \lim_{x\to2} \frac{2 x^2-3 x-2}{x-2}\)
  2. \(\displaystyle \lim_{x\to\infty} \frac{2 x^2-3 x-2}{x-2}\)
  3. \(\displaystyle \lim_{x\to\infty} \frac{2 x^2-3 x-2}{x^2-2}\)
  4. \(\displaystyle \lim_{x\to2} \frac{x+1}{x^2-4}\)
  5. \(\displaystyle \lim_{x\to4} \frac{x+1}{(x-4)^2}\)
  1. Let \(f(x) = x^3 - x^2 - x - 1\). Write a complete sentence explaining why there is a number \(c\in(0,2)\) such that \(f(c) = 0\).
Figure 1: The graph of \((x-1)/(x^3 - x^2 + x - 1)\)
Figure 2: Figure for limits and continuity