Mathematics

Session Functions, Limits and Continuity -3

Session Objectives  Limit at Infinity  Continuity at a Point  Continuity Over an Open/Closed Interval  Sum, Product and Quotient of Continuous Functions  Continuity of Special Functions

Limit at Infinity A GEOMETRIC EXAMPLE: Let's look at a polygon inscribed in a circle... If we increase the number of sides of the polygon, what can you say about the polygon with respect to the circle? As the number of sides of the polygon increase, the polygon is getting closer and closer to becoming the circle! If we refer to the polygon as an n-gon, where n is the number of sides, Then we can write

Limit at Infinity (Cont.) The n-gon never really gets to be the circle, but it will get very close! So close, in fact, that, for all practical purposes, it may as well be the circle. That's what limits are all about!

Limit at Infinity (Cont.) A GRAPHICAL EXAMPLE: Now, let's look at the graph of f(x)=1/x and see what happens! Let's look at the blue arrow first. As x gets really, really big, the graph gets closer and closer to the x-axis which has a height of 0. So, as x approaches infinity, f(x) is approaching 0. This is called a limit at infinity.

Limit at Infinity (Cont.) Now let's look at the green arrow... What is happening to the graph as x gets really, really small? Yes, the graph is again getting closer and closer to the x-axis (which is 0.) It's just coming in from below this time.

Some Results

Example - 1 Solution :

Example – 2 Solution :

Example - 3 Solution :

Solution Cont.

Example – 4 Solution :

Solution (Cont.)

Continuity at a Point Let f(x) be a real function and let x = a be any point in its domain. Then f(x) is said to be continuous at x = a, if If f(x) is not continuous at x = a, then it is said to be discontinuous at x = a.

Left and Right Continuity f(x) is said to be left continuous at x = a if f(x) is said to be right continuous at x = a if

Continuity Over an Open/Closed Interval f(x) is said to be continuous on (a, b) if f(x) is continuous at every point on (a, b). f(x) is said to be continuous on [a, b] if

Sum, Product and Quotient of Continuous Functions Let f and g be continuous at x = a, and let be a real number, then

Continuity of Special Functions (1) A polynomial function is continuous everywhere. (2) Trigonometric functions are continuous in their respective domains. (5) Inverse trigonometric functions are continuous in their domains. (4) The logarithmic function is continuous in its domain. (6) The composition of two continuous functions is a continuous function.

Example – 5

Solution (Cont.) So, f(x) is continuous at x = 0.

Example –6

Solution (Cont.) So, f(x) is continuous at x = 0.

Example – 7

Solution (Cont.) So, f(x) is discontinuous at x = 0.

Example – 8

Solution (Cont.)

Example –9

Solution (Cont.)

Example –10

Solution (Cont.)

Thank you