2.1 Rates of Change & Limits 2.2 Limits involving Infinity Intuitive Discussion of Limit Properties Behavior of Infinite Limits Infinite Limits & Graphs of Rationals End Behavior Models

Graph Using your tables, set x = -0.5 and step 0.1 then determine the value of y as you near 0 Go to the graph and trace, does it agree? What is the problem here? What should the value be? Now do the same process with x = 100 and the step = 50.

Define a Limit When we take the limit of a function we are looking at the behavior of a function as it nears a certain value. We examine the ‘behavior’ from the left and from the right of the value, called ‘c’. In our example, what did we determine was the ‘limit’ as x got close to 0, close to infinity? Can we have a limit and not have a functional value? Could we have a limit and a functional value be the same? Could we have a limit and a functional value not the same?

Examine Limits & Functional Values Graph the following function on your calculator: Trace the function as you get close to x = 2. What is the limit as x approaches a value of 2?

Suppose the graph looked like this. Does the limit change? What is the functional value? How can we redefine the function y such that the functional value and the limit is the same value?

Properties of Limits 1.Sum and difference: the limit of a sum or difference is the sum or difference of the individual limits. 2.Product : the limit of a product is the product of the limits 3.Constant Multiple: the limit of c times a function is the limit of the function then times c. 4.Quotient: the limit of a quotient is the quotient of the limits. 5.Power rule: the limit of a function to a power, rational or integer power, is the limit of the function then raised to a power. 6.Constant : the limit of a constant is the constant. 7.Polynomial rule: the limit of a polynomial as x approaches c is f (c). 8.Rational function: the limit of a rational function, where both numerator and denominator are polynomials, is the limit of the numerator divided by the denominator.

Examples

Sandwich Theorem (Squeeze Theorem) Given the following conditions and looking at a graph we have: L c

Using the Sandwich Theorem Show by using the Sandwich Theorem the following:

Free Fall Problem Define Average Speed: Define Instantaneous Speed: Speed at the instant when t = c. On the interval of time t = a to t = b

As h approaches 0, the average speed becomes the instantaneous speed. Instantaneous speed is a value L and is called the limiting value of the average speed. Given the function : Determine the average speed on the interval t = 0 to 5. Determine the instantaneous speed at t = 5.

Limits Involving Infinity

Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of y = f(x) if the limit as x approaches + or – infinity is b. In other words, the behavior of the function shows the y values approaching b as x gets very large or very small. Graph: Use an x window from –10 to 10. Go to your tables and start at x = 0 and increase by 1, what happens as x gets very large; repeat for x getting very small. What are the limiting values?

Graph: Again use the tables, starting at 100 and increasing by 100. What is the limit as x gets infinitely large? Could we use the Sandwich Theorem to prove this analytically? Using this conclusion, determine the following limit algebraically:

Vertical Asymptote Determine the following limits by examining the graph again. The line x = a is a vertical asymptote of the graph of y = f(x) if either of the limits as x approaches a from the right or the left has a behavior suggesting infinite values.

End Behavior Models Functions may differ for small values of x, but as x approaches positive or negative infinity, they are identical for large absolute values of x. Behavior models need not be the same as either of the two functions being examined. The effect of the largest power of x in any rational function determines the end behavior model. To determine the end behavior model of a rational function: use the largest powers of x and take the limit as x approaches infinity.

Find the end behavior model of the function:

Given the function f(x), determine the right-end and left-end behavior models by graphing and zooming out several times. What function does this model in the extreme values of x? Check your answer by using limits.

Find an end behavior model and check your answer on the calculator. What happens when we algebraically determine the limit as X approaches infinity? What characteristic can we look for to determine if the function has a horizontal asymptote or end behavior acting like a polynomial function?

Investigate y = f(x) as x approaches + or – infinity by using a substitution such as 1/x approaching 0. Is equivalent to this statement

Discuss asymptotes in the following problems: