MthT 430 Projects Chapter 6a Limits and Continuity
MthT 430 Projects Chapter 6a Limits and Continuity
Limits
1.
(Omit for now - will discuss after MidTerm Assessment) Let f(x) be a function such that
•
domain( f) = [0,1).
•
For all x (in [0,1)), 0 ≤ f(x) < 1.
•
The function f is increasing on [0,1).
Show that there is a number L, 0 ≤ L ≤ 1, such that
lim
x → 1
−
f(x) = L.
Hint:
Construct a binary expansion for L.
2.
Discuss the continuity of the function described on p. 97 and whose graph is sketched in FIGURE 14.
3.
Prove: If g is continuous at a, g(a) ≠ 0, then there is a δ > 0 for which (a − δ,a + δ) is contained in the domain of [1/g].
4.
Spivak, Chapter 6, Problem 3.
5.
Spivak, Chapter 6, Problem 13.
6.
If f is continuous at 0, f(0) = 0, g(x) is defined for all x near 0, and |g| is a bounded function, say all you can about
lim
x → 0
f(x) ·g(x).
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On 18 Sep 2014, 21:17.