Mathematics 133 – Intensive Calculus for Science 1 Discussion 3 – More on Limits September 30, 2005

Background

Recall that we have said that if f(x) is a function defined on some interval containing x = c, except possibly at x = c, then lim_{x→c} f(x) = L if we can make the values f(x) as close as we like to L by taking x sufficiently close to c (but not equal to c). Today we want to gain more experience working with this idea, considering limits graphically and numerically. Similarly, we can define the “one-sided” limits lim_{x→c^{+}} f(x) and lim_{x→c^{−}} f(x), where we take x sufficiently close to c but only greater than c in the first case, and only less than c in the second. We also know that saying f is continuous at x = c should mean lim_{x→c} f(x) = f(c).

Discussion Questions

A) The following graph shows y = f(x).

1) What are lim_{x→1^{+}} f(x) and lim_{x→1^{−}} f(x)?

2) Do you think lim_{x→1} f(x) exists? Why or why not?

3) Is f continuous at x = 1? Why or why not?

4) What are lim_{x→3^{+}} f(x) and lim_{x→3^{−}} f(x)?

5) Do you think lim_{x→3} f(x) exists? Why or why not?

6) Is f continuous at x = 3? Why or why not?

7) What are lim_{x→5^{+}} f(x) and lim_{x→5^{−}} f(x)?

8) Do you think lim_{x→5} f(x) exists? Why or why not?

9) Is f continuous at x = 5? Why or why not?

10) In your own words, explain what must be true about the one-sided limits at x = c in order for f to be continuous at c.

B) You will need a calculator for this question. Let f(x) = \frac{sin(3x)}{x}.

1) Make a table of values for f(x) at x = −.1, −.01, −.001, −.0001, .0001, .001, .01, .1.

2) Make a conjecture (that is an “educated guess”) about lim_{x→0} \frac{sin(3x)}{x} based on the values in your table.

3) What does the graph y = \frac{sin(3x)}{x} look like for x ∈ [−2, 2]. Does this confirm your answer in 2? Explain.

Please hand in your work to Prof. Ballantine at the end of class.