MA1102 Calculus Tutorial 4

Topics Covered:

(Grappling with infinity). Limits at infinity and infinite limits. Asymptotes. The Squeeze Theorem. Continuity at a point. Continuous function. The Intermediate Value Theorem.

Textbook: Chapter 4, 4.5 -

Typically the tutorial will take you to the following conclusion.

1. If and f (x) > 0 ( f (x) < 0 ) for x ¹ a and near a, then

.

2. If , then .

(Similar result is true with limit replaced by one-sided limit.) The serious minded should provide a proof for the above results. Behaviour of a function for large x is synonymous with asymptote. Think of limit at infinity as how we like to think the function is like for large x. We restrict attention to when the function for large x is like a constant. You can easily extend the behaviour at infinity to include a straight line function (oblique asymptote). Vertical asymptote is exactly the opposite, it is the vertical line at the point where the limit (left or right limit) is ± ¥ . The Intermediate Value theorem is a deep theorem. There are some ingredients in the theorem that are unsaid. Firstly, the interval is a 'connected ' piece. the closed and bounded interval is a connected piece with both ends. A consequence of this theorem is that the range is also a connected piece with both ends. The graph of a continuous function on a closed and bounded interval is like a 'deformed wire' . The common application is to finding a root. For this you will require two points in the domain with images of opposite signs.