Agenda - Tuesday, August 7th Check folders finish discussing syllabus & notebook guidelines notes on rates of change & limits hw: understanding the limit (1 – 9) due 08/08

Rates of Change and Limits 2.1 Rates of Change and Limits

Definition: Limit of a Function The limit notation is This is said “the limit of ,as x approaches a, equals L” This means that as the distance between x and a becomes smaller, the value gets closer and closer to L. x need not ever equal a for the limit to exist.

Fuzzy Definition of Limit

Precise Definition of a Limit

How do we evaluate this concept of limit? 1.1 1.01 1.001 1.0001 1.00001 undefined 1 0.99999 0.9999 0.999 0.99 0.9 How do we evaluate this concept of limit? Example: Answer: From the table we can see that the limit is 0.5 or 1/2

1.1 1.01 1.001 1.0001 1.00001 2 1 0.99999 0.9999 0.999 0.99 0.9 Note: this is the same function as in the last example except at x = 1, so we will get the same table of values.

1.1 1.01 1.001 1.0001 1.00001 2 1 0.99999 0.9999 0.999 0.99 0.9 As x approaches 1, the values of g(x) approach 0.5. The fact that g(1) = 2 does not matter when evaluating the limit

Guess the value of . Example The function f(x) = (sin x)/x is not defined when x = 0. Using a calculator, we can make a table of values to estimate the value of the limit

From the table and the graph, we guess that Example From the table and the graph, we guess that We will prove this guess later in the course.

Special Cases The value of f(x) does not exist at a. The value of f(x) exists but is a point off the curve.

THEOREM 1 PROPERTIES OF LIMITS 1.) Sum Rule: 2.) Difference Rule:

Limit Laws 3.) Product Rule: 4.) Constant Multiple Rule:

Limit Laws 5.) Quotient Rule:

Example Evaluate the given limits.

Evaluate the given limits: By the sum law a.)

By the difference law b.)

By the constant multiplier law

By the product law d.)

e.)

Important Note! When applying the quotient law, the limit of the denominator function must be nonzero. If the limit of the denominator function is zero, then the quotient rule does not apply. Example We will learn to deal With this case later

Properties of Limits The limit of a sum is the sum of the limits. The limit of a difference is the difference of the limits. The limit of a product is the product of the limits. The limit of a constant times a function is the constant times the limit of the function. 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0) Show Properties of Limit transparency

More Limit Laws 6.) Power Rule: 7.) Constant Function Limit: (This is obvious, since a constant function will only approach the constant that defines it.)

8.) Identity Function Limit: 9.) Power Function Limit: (This is seen by applying the power law to the identity function limit)

10.) Root Function Limit: 11.) Root Rule:

Example This procedure of breaking the limit up term by term using the sum and difference laws, then pulling out coefficients with the constant multiplier law, then using the power, identity, and constant function limits can be used for any polynomial. By examining the next to last line of the solution we can see that the limit as x->a of the polynomial is just the polynomial evaluated at a. This motivates the following theorem.

THEOREM 2 Direct Substitution Theorem The easiest method for finding limits is direct substitution. Direct substitution is simply an algebra evaluation. Use Plug and Chug transparency

Example Quotient Law Direct Substitution Theorem

Example Notice that the quotient law is not applicable here, since the denominator function has a limit of 0 as In order to evaluate this limit we need to take another approach. Aside

Back to our original problem

What To Do When Direct Substitution Won’t work Problems arise when using direct substitution such as: Creating zero in the denominator when substituting. In such cases, you will need to employ algebraic manipulations before doing the direct substitutions.

Strategies for Finding Limits Reduce a fraction Rationalize a denominator Rationalize a numerator Divide by the monomial Turn a fraction into a product of two fractions Simplify a fraction Use long division

Strategies for Finding Limits cont. 8. Use trig substitutions 9. Use the squeeze theorem

The Sandwich Theorem If when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, and then

Example Evaluate We saw that the did not exist because of the infinite number of oscillations near the y-axis, so using the product law will not work. Consider the following. .

One Sided Limits Left sided limits are found by approaching a from the left (moving from -∞ to a) and are written as Right sided limits are found by approaching a from the right (moving from +∞ to a) and are written as

One Sided Limits Example

What is the limit as x approaches -2 from the left? So the left sided limit is 1, or

What is the limit as x approaches -2 from the right? So the right sided limit is -4, or

So we know the left and right side limits of g(x), but what about the ordinary limit?

When Limits Do NOT Exist Limits do NOT exist if: the left-hand limit (approaching from the left) does not equal the right-hand limit (approaching from the right). So in our previous example Does NOT Exist or (DNE)

When Limits Do NOT Exist Limits do NOT exist if: (There are 3 cases) 1. the left-hand limit (approaching from the left) does not equal the right-hand limit (approaching from the right). Example: 2. The limit, L, approaches +∞ or -∞ Example 3. The function is oscillating wildly at a.

Investigate . Again, the function of f(x) = sin ( /x) is undefined at 0.

Evaluating the function for some small values of x, we get: Similarly, f(0.001) = f(0.0001) = 0.

On the basis of this information, we might be tempted to guess that . This time, however, our guess is wrong. Although f(1/n) = sin n = 0 for any integer n, it is also true that f(x) = 1 for infinitely many values of x that approach 0.

The graph of f is given in the figure. The dashed lines near the y-axis indicate that the values of sin( /x) oscillate between 1 and –1 infinitely as x approaches 0.

Since the values of f(x) do not approach a fixed number as x approaches 0, does not exist.

Theorem 3 One-sided and Two-sided Limits if and only if and

DNE Example Let the graph to the right represent the function h(x) Estimate the following limits: a.) d.) b.) e.) DNE c.) f.)

Example Remember that the definition of the absolute value function is

This means that it has a different definition to the right of 0 from that to the left of 0. Keep in mind that we had a Theorem saying the left and right side limits must equal for the limit to exist