On Monday we wrote down the Calculus of Limits. We stated that parts (iii) and (iv) have nasty proofs and left them on the webpage. We proved parts (i) and (ii) and showed in part (v) that if the limit exists, it must have that form. We showed that for any polynomial $p(x)$, that $\lim_{x\rightarrow a}=p(a)$. Finally we proved the special limit:
$\lim_{x\rightarrow 0}\frac{\sin x}{x}=1$
In the tutorial we ran out of questions fairly early, so I threw the exercises up on the projector and allowed ye work away. Ye put up your hand if ye wanted assistence.
On Wednesday we considered infinite limits – when does a function tend to infinity and what does a function do as $x$ tends to infinity. Also we gave the crucial definition of a continuous function.
Finally I stated that the Sample Test will be up before or on Wednesday 20 – the test is on 27/10/10. Question 1 will be from the exercises, Question 2 from past exam papers, and Question 3 will be about definitions.

Problems

You need to do exercises – all of the following you should be able to attempt. Do as many as you can/ want in the following order of most beneficial:

Wills’ Exercise Sheets

Q. 1\(ix), 2,3,4,5 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise2.pdf

More exercise sheets

Section 2 from Problems

Past Exam Papers

Q. 1(b), 3(a) from  http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/MS2001Sum2010.pdf

Q . 1(b), 2(b) from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/MS2001s09.pdf

Q. 1(b), 3(a) from http://booleweb.ucc.ie/ExamPapers/exams2008/Maths_Stds/MS2001Sum08.pdf

Q. 1(b), 2(a)(ii-iii), 3(a) from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 2(b), 3(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 2(b), 3(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

Q. 1(b), 3(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 1(b), 2(a) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 1(a), 2(a), 3 from http://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf

From the Class

1. Prove the following proposition:

Suppose that $f,g$ are functions for which $f(x)=g(x)$ for all $x \neq a$. If $\lim_{x\rightarrow a}f(x)$ exists then so does $\lim_{x\rightarrow a}g(x)$ and moreover they are equal.

2. Investigate, for $n$ an odd natural number,

$\lim_{x\rightarrow -\infty}x^n$

3. Investigate

$\lim_{x\rightarrow -2}\frac{x-3}{x^2+3x+2}$