MA1102 Calculus Tutorial 3

Topics Covered:

Intervals (Notation). Limit at a point. Left and right limits. e -d definitions. Rational, odd and even functions. Piecewisely defined functions. Functions having no limits at a point.

Textbook: Chapter 4, 4.1-4.4

Limit is central to calculus. Without doubt it is the most important concept to grasp. Think of distances as you recall the e -d definitions. You can rephrase definitions in terms of distances, if it helps. Be prepared to keep going back to the definition of limit in one form or another for understanding. You will not be asked to provide proofs using the definition. But it would be very rewarding, if you can try to work out the proofs for the limits of sum and product of functions. It is not necessary to memorise the definition but you should have a good idea why the definition is in its present form. For the curious and the serious minded, provide the proof for the limit of the quotient of two functions. What does it mean to say a function has no limit at a point? How does one go about showing this? Questions like these are equally important. Polynomial functions are popular, next in line is rational functions. The factor and remainder theorem is useful here. What does cancellation of factors here mean? How can it help you to compute limits? For this matter the following identity is useful in removing n-th root and consequently helps in cancellation.

a ⁿ - b ⁿ = ( a - b) (a ^{n - 1}+ a ^{n - 2}b + a ^{n - 3}b ² + ... + a b ^{n - 2} + b ^{n - 1})