MODULE 2

LIMITS AND CONTINUITY

Limit of a Function:

Note: x can approach a form the left or from the right.

Example 1:

It is not defined at x = 2.

So the limit will be 4. In this case, for example, x can approach 2 from the left by taking values

.9, .99, .999, .9999, .99999, …….

or from the right by taking values

2.1, 2.01, 2.001, 2.0001, …….

and the common value attained by the function is 4.

Example 2:

Answer: 1

x	0.1	0.01	0.001	0.0001
	0.9983342	0.9999833	0.9999998	1.0

Note: The function is not defined at x = 0.

Note: If p(x) is a polynomial in x then

Thus every polynomial function is a continuous function for all real numbers.

Infinite Limit: If a function decreases without bound as x approaches a, we write

If a function increases without bound as x approaches a, we write

For example:

Find the following limits a) graphically b) by using a table of values ( as done in Example 2)

Does the limit of the function exist as x approaches 2?

Note:

Verify this graphically.

Note: You may to use the fact that

The answer to #6 is 2. Verify it graphically using trace and zooming features of a graphing calculator.

Continuity of a Function at a Point:

We say that a function f is continuous at a point x = a if

f (a) is defined;
.

A function that is not continuous at a is said to have a discontinuity at the point a.

A function is continuous on an interval if it is continuous at every point of the interval.

Exercises: Find the points of discontinuity of the following functions:

Note: Find the domain first and use the graph of the function to guess the answer.

7. Define m(2) so that m(x) is continuous at x = 2.

Exercises:

Find each point of discontinuity.
Which of the discontinuities are removable? Not removable?

_{Using the graphing
calculator we get the following graphs:}

_{For #9, type
y = (x+2)(x<1)+(-x+1)(x>1)}

_{Its domain is
|x| < 3 and so the function is continuous for –3 < x < 3.}

Horizontal Asymptote:

The line y = b is a horizontal asymptote of the graph of the function y = f(x) if either

Vertical Asymptote:

The line x = a is a vertical asymptote of the graph of the function y = f(x) if either

Examples:

Let y = 2 + 3/(x-1)

and so y = 2 is a horizontal asymptote

and so x = 1 is a vertical asymptote.

End Behavior Models:

The function g is a right end behavior model for f if and only if

The function g is a left end behavior model for f if and only if

Examples:

Intermediate Value Property of a Continuous Function:

Let f be a continuous function on the closed interval [a , b]. Then f(x) assumes every value between f(a) and f(b) i.e. if l is number between f(a) and f(b), then there is at least one number c in the open interval (a , b) such that f(c) = l.

Example:

Note that f(0) = -1 and f(1) = 1.

Average Rate of Change of a Function:

The average rate of change of a function f on the interval [a,x] is defined by

Examples:

Slope of a Tangent Line to the curve y = f(x) at x = a:

Let P(a , f(a)) be a point on the curve y = f(x). Let Q(a+h, f(a+h)) be a point on y = f(x). The slope of the curve y = f(x) at the point P(a , f(a)) is the number