Write-ups of the problems below are due Monday Oct. 9

Monday Oct 2

We did: A.2: proved Theorems 1,4iii.  We will take Theorem 5 for granted.

Solve, write up and turn in:  In Sec. 2.5 problems 1, 2, 5(explain here why you are allowed to simplify the fraction when calculating the limit), 9, 17
(You do not need to use the epsilon-delta definition of limits,
use the properties of limits. We have worked so hard to prove them...now we can use them!)

Bonus problem (extra 3 points) Prove problem 3a from Friday Sept. 28  by induction on the degree of the polynomial.
Hint: write polynomials of degree n in the form P(x)=cnx^n+cn-1x^(n-1)+...+c1x+c0.
and note that P(x)=cnx^n+polynomial of degree (n-1).
Tuesday Oct 3

Plan: Sec. 2.5:  sin x/x ->1 (as x->0), lateral limits, cases when limits do not exist. A.2: Theorem 6.

Solve, write up and turn in:
Sec. 2.5  2, 4, 8, 10, 14, 15, 19a,g,f,  20 c,d,g

Wednesday Oct 4

We did today: we calculated limits, we proved Theorem 6 of A2 (the squeeze theorem) and discussed applications, we defined lim as x->infinity.

Solve, write up and turn in: Sec. 2.5 (revisit yesterday's problems and) 18 c,d,e,f and also:
1. a) Does the function  cos x  have a limit as  x->infinity? Explain.
b) Does the function  cos (1/x)  have a limit as  x->0? Explain.
c) Show that the limit of  xcos (1/x) as x->0  is 0.
2. Let f(x) be a bounded function  (this means that there are some numbers b,B so that b<f(x)<B for all x).
Show that the limit of the function xf(x) as x->0 exists and equals 0.

Bonus problem (extra 3p.): Write down the definition of "the limit of f(x) as x goes to  negative infinity equals L".

Note: For Tuesday's assignment I wrote by mistake Sec. 2.6 instead of 2.5.

Thursday Oct 5

Topics: Continuity Sec. 2.6 and  A.2 Theorems 7,8,9,10. We defined composition.

Solve:
1a,b,e

Solve, write up and turn in:
Sec. 2.6 (page 79)1 f,g,h  and
3. Consider the function f(x)=sin(3x)/x if x is not 0 and f(0)=C.
For which number C the function f(x) is continuous on the whole real line?

Bonus (extra 5 points) Consider the function f(x)=0 if x is rational and
=1 if x is irrational.
Show that this function is discontinuous at all the points.
At which points is  xf(x)   continuous?
Hint: please read first the Remarks at the bottom of page 5 of Bounds. The implication is that:
any real number can be approached through rationals only, or through irrationals only.

Friday Oct 6

Topics: Sec. A2 Theorem 11. Continuous functions on closed intervals - Sec. A.3: Theorem 1 (no proof required, but,
attention math majors and minors: please do read the proof), Theorem 2.

Solve, write up and turn in: 4. Write the following functions as a composition of two simpler functions:

Also: Sec. 2.6 (page 79/80) 1 e (and explain which theorems you used and how), and also
*15 c, 16 b,c,  21, 22, 29 to all of which answering the additional questions:
explain (if it is the case) if max/min necessarily exists,
and if those functions have a supremum, or infimum,
on the stated intervals, explain why and what are their values *