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Class Note for ECON 2370 at UH

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Date Created: 02/06/15
Math 1314 Lesson 1 Limits What is calculus The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 183911 centuIy 1 How can we nd the line tangent to a curve at a given point on the curve 2 How can we nd the area ofa region bounded by an arbitraIy curve The investigation of each ofthese questions relies on the process of nding a limit so we ll start by informally de ning a limit and follow that by learning techniques for nding limits Limits Finding a limit amounts to answering the following question What is happening to the y Value of a function as the xValue approaches a speci c target number If the y Value is approaching a speci c number then we can state the limit of the function as x gets close to the target number Example 1 7K y 9074 2 0 7 o 60 50 40quot l 30 20 139 f 10 1 9 8 7 6 s 432 i10 1 2 3 4 5 a 7 8 9391 720 4 0 5 o 76 0 70 3 or 790 mm It does not matter Whether or not the x value every reaches the target number It might or it might not Example 2 When can a limit fail to exist We will look at two cases Where a limit fails to exist note there are more but some are beyond the scope of this course Case 1 The y value approaches one number from numbers smaller than the target number and it approaches a second number from numbers larger than the target number Case 2 At the target number for the x value the graph of the function has a vertical asymptote 79724757574444 34557291 For either of these two cases we would say that the limit as x approaches the target number does not exist Definition We say that a function f has limit L as x approaches the target number a written limfx L ifthe value x can be made as close to the numberL as we please by taking x suf ciently close to but not equal to a Note that L is a single real number Evaluating Limits There are several methods for evaluating limits We will discuss these three 1 substituting 2 factoring and reducing 3 nding limits at in nity To use the rst two of these methods we will need to apply several properties of limits Properties of limits Suppose lim fx L and limgx M Then 1 liiraifx fx L for any real number r cfx fx CL for any real number c 13mm gx 13mm 933m L M gigIfxgx 133 fx1xi algx 1M Emmzitlzm L 5 providedM 0 Ha gx 11mgx M We ll use these properties to evaluate limits 59 Substitution Example 3 Evaluate lim3x2 4x 5 xgt2 3 Example 4 Evaluate 11m Ho x 1 3 Example 5 Evaluate lim 3x2 xgt4 k What do you do when subst1tut10n g1ves you a value 1n the form 6 where k 15 any nonzero real number 2 5 Example 6 Evaluate 11m x H3 x3 Indeterminate Forms 0 What do you do when subst1tutlon glves you the value 6 This is called an indeterminate form It means that you are not done with the problem You must try another method for evaluating the limit See if you can factor the function If you can you may be able to reduce the fraction and then substitute 2 4 5 Example 7 Evaluate llm Xgtl x 2 5 6 Example 8 Evaluate llm Xgt2 x So far we have looked at problems Where the target number is a speci c real number Sometimes we are interested in nding out What happens to our function as x increases or decreases Without bound Limits at In nity 2x 2 2 Example 9 Consider the function f x 1 x What happens to x as we let the value of x get larger and larger x 10 50 100 1000 10000100000100000010000000 fix We say that a function x has the limitL as x increases Without bound or as x approaches in nity Written lim f x L if x can be made arbitrarily close toL by taking x large enough We say that a function x has the limitL as x decreases Without bound or as x approaches negative in nity Written lim f x L if x can be made arbitrauly close to L by taking x to be negative and suf ciently large in absolute value We can also nd a limit at in nity by looking at the graph of a function Example 10 Evaluate lim 2x7 7 Hm x 7m 712 72 74 4 2 12 16 2 We can also find limits at in nity algebraically or by recognizing the end behavior of a polynomial function Example 11 Evaluate lim4x3 7x 5 xgtoo Limits at in nity problems often involve rational expressions fractions The technique we can use to evaluate limits at infinity is to divide every term in the numerator and the denominator of the rational expression by xquot where n is the highest power of x present in the denominator of the expression Then we can apply this theorem 1 n Theorem Suppose n gt 0 Then lim 0 and lim n 0 provided is defined 90 x raise x x After applying this limit we can determine what the answer should be YOUMUST KNOW THIS PROCEDURE 2 2 5 1 Example 12 Evaluate limxz x H 3x 2x 7 Often students prefer to just learn some rules for nding limits at in nity The highest power of the variable in a polynomial is called the degree of the polynomial We can compare the degree of the numerator with the degree of the denominator and come up with some generalizations o If the degree of the numerator is smaller than the degree of the denominator then lim 0 Hoe gx o If the degree of the numerator is the same as the degree of the denominator then you can nd lim by making a fraction from the leading coef cients of the Hoe g x numerator and denominator and then reducing to lowest terms 0 If the degree of the numerator is larger than the degree of the denominator then it s best to work the problem by dividing each term by the highest power of x in the denominator and simplifying You can then decide if the function approaches 00 or oo depending on the relative powers and the coef cients The notation lim f x 00 indicates that as the value of x increases the value of the xgt o function increases without bound This limit does not eXist but the 00 notation is more descriptive so we will use it 2 4 5 4 Example 13 Evaluate llme x H x x l 5 2 3 4 Example 14 Evaluate llme x H 4x 2x 8 4 5 Example 15 Evaluate limzx H x 9x 9 From this lesson you should be able to State the two basic problems of the calculus De ne limit indeterminate form Find a limit as x approaches a target number from a graph off State when a limit fails to exist k Evaluate 11m1ts where subst1tution gives 6 k 72 0 Evaluate limits by substitution or by factoring Evaluate limits at in nity

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