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Page 14 When Can We Plug It in and When We Cant? Are there any cases we cant just plug it in? Yes and No. Yessometimes the simply-plug-it-in method will not give the answer desired. Nosince we still need to plug it in first before we can tell whether the answer weve got is desired or not. Examples: 1. lim x 2 (x 2 + 1) = 2 2 + 1 = 5 (Yes! 5 is the desired answer) 2. lim x 0 1/x 2 = 1/0 2 = 1/0 = undefined (But undefined is not the desired answer) 3. lim x 2 (x – 2)/(x 2 – 4) = (2 – 2)/(2 2 – 4) = 0/0 (But 0/0 is not the desired answer) Are we saying 1/0, 0/0 and undefined are not acceptable answers in evaluating limits? Yes. What do we do then? Re-evaluate the limits using other means. First, we will show you why you have to re-evaluate the limit whenever you have 1/0 (and in general, k/0, where k 0). Recall in limits, we have to use numbers close to a (the number x is approaching), we cant really use a. We only can use a (i.e., directly plug a into the function f) if f is continuous at a. If you look back Examples 2 and 3, 0 is the point of discontinuity (POD) of 1/x 2 and 2 is the POD of (x – 2)/(x 2 – 4).

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Page 15 What is k/0 (k 0)? We know, when k 0, k/0 is undefined. However, in terms of limit, its really either a positive infinity (+ ) or a negative infinity (– ). As you can see from these two tables: Of course, depends on whether k is positive or negative and, also, whether 0 is positive or negative. If k is positive and 0 is positive, then k/0 = ________. If k is positive and 0 is negative, then k/0 = ________. If k is negative and 0 is positive, then k/0 = ________. If k is negative and 0 is negative, then k/0 = ________. x.1.01.001.0001.000010 1/x x–.1–.01–.001–.0001–.000010 1/x

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Page 16 Positive and Negative Zero? Examples: 1.2.3. Definition of Positive Zero and Negative Zero: Positive zero (denoted by +0) is defined as a quantity very close to 0 but slightly greater than 0 (i.e, positive). Negative zero (denoted by –0) is defined as a quantity very close to 0 but slightly less than 0 (i.e, negative). 4.5.6. 7.8.9. Note: Usually you will be given problems like #3, #6 and #9 and you will not be given the graph. Therefore, if the limit is of the form k/0 (i.e., a non-zero number divided by zero), you must consider the one-sided limits first and then make a conclusion on the (both-sided) limit.

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Page 17 What Happen if the Numerator Is also 0? What happen if k = 0, i.e., 0/0? Well, it is a special case on its ownso special that it has a nameindeterminate form of [0/0]. Definition of Indeterminate Form: A limit is an indeterminate form if, by direct substitution, its one of the following cases: [0/0], [ – ], [ /,] [0 ], [1 ], [ 0 ] and [0 0 ]. The following is one example of each: Note: We will talk about [0/0] next and [ – ] later in this presentation. [ / ] and [0 ] will be discussed later in this course while the last three forms you have to wait until Course II.

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Page 18 Indeterminate Form of [0/0] The following are some examples of indeterminate form of [0/0]: As you can see, each of the above is [0/0] when we just plug the number into the function. Examples 1 and 2 are most common because the functions are a family of functions called ________ functions. After you know its [0/0], you can (and you must) re-evaluate the limit by simplifying the rational function (i.e., factor the numerator and/or denominator and cancel out the common factor) and re-plugging the number in the simplified version. Note: In problems of the above type (i.e., the function is rational and its of the form [0/0]), knowing how to factor is the key to solve these problems. Some problems are harder to factor than the others (for example, problem c is harder than problem a or b), but all of them should be factorable, thats because both numerator and denominator contain a common factor that can be cancelled. We call this common factor a problematic factorthe one that causing the both numerator and denominator to be 0. However, once it gets cancelled out, the re-plug-it-in will reveal what the (true) limit is. And one more hint: if x is approaching a, this common (or problematic) factor should be ______.

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Page 19 Indeterminate Form of [0/0] (contd) The function in example 3 of last page involves radical, in particular, square root. We are going to divide this type of problems in three cases: 1. If the denominator has a square root, then _______________________________________. 2. If the numerator has a square root, then _________________________________________. 3. If the denominator and the numerator has a square root, then ________________________. Note: Even though radicals can mean cube roots, 4 th roots, etc., we only show you how to deal with square roots because i. most problems of this type involve square roots only, and ii. the ones with the cube roots (or higher roots) are much more difficult to do. The function in example 4 of last page is a complex fraction. If its [0/0], to evaluate the limit, one must simplify the complex fraction first, i.e., reduce it to a regular fraction. Again, most of these problems need basic and sometimes clever algebraic manipulation.

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3.6 Solving Quadratic Equations

3.6 Solving Quadratic Equations

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