Math 141 Homework #3
Due Tuesday, 9/4/07
Extra Problems
#1. Suppose that f(x) = p(x)/q(x) is a rational function, where p(x) and q(x) are polynomials.
When does lim
x→0f(x) = 0? When is lim
x→0f(x) a nonzero real number? When does lim
x→0f(x) not exist?
Your answer should be fully explained, and should cover rational functions as possibilities. That is, I should
be able to take any rational function at all and use your answer to decide the value of its limit as x→0.
(So, for example, it is not sufficient to only give an example of each of the three cases.)
#2. On Tuesday 8/28 in class, we examined a function s(x) that swaps the first two digits of the decimal
expansion of x. What about the function r(x) that swaps the first and third digits of x(so, e.g., r(1.23456) =
1.43256, r(2.121212) = 2.121212)? For which real values of ais rcontinuous at a?
Bonus problem: Let T(x) be the function with domain (0,∞) defined as follows:
•T(a/b) = 1/b, if a/b is a fraction in lowest terms;
•T(x) = 0, if xis an irrational number.
So, e.g., T(1/2) = 1/2, T(0.375) = 1/8 (because 0.375 = 3/8), T(π) = 0.
For which values of ais T(x) continuous at a?
