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# Class Note for MATH 1314 with Professor Marks at UH

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Date Created: 02/06/15
Math 1314 Online Week 1 Notes Instructor Marjorie Marks Email mmarksc mathuhedu Course Website onlinemathuheducourses Click on the link to Math l3 l4 CASA Website casauhedu My personal website mathuheduNmmarksc For Course Orientation see the Orientation video posted under Week 1 at onlinemathuheducourses on the 1314 page Open the Popper titled popl at casauhedu Section 104 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 century 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 arbitrary curve The investigation of each of these 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 Informal definition Finding a limit amounts to answering the following question What is happening to the yvalue of a function as the x value 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 nurnber Look at these graphs and nd the limit as X gets really close to l in both cases y Example 1 Find llIIZI fx Find ling fx It does not matter whether or not the x value every reaches the target number It might or it might not Example 2 Find lim fx xgt1 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 xvalue the graph of the function has a vertical asymptote For either of these two cases we would say that the limit as x approaches the target number does not exist More Formal De nition We say that a function f has limit L as x approaches the target number a written lim f x L Ha if the value x can be made as close to the number L as we please by taking x sufficiently 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 finding limits at infinity To use the first two of these methods we will need to apply several properties of limits Properties of limits Suppose 1imfx L and limgx M Then 59 1iinfx39 lini fx39 L for any real number r cfx clilnafOc CL for any real number c 1331ku ggx 133 1 00 i 133 gx L M Iguana 1333 1 00 133 gx 1M mm L f i providedM 2 0 Ha gx 11mgx M We ll use these properties to evaluate limits Substitution Example 3 Evaluate 1imx2 6x 5 xgt3 2 5 Example 4 Evaluate 11m x H4 3 2x k What do you do when subst1tut1on g1ves you a value 1n the form 6 where k 15 any non zero real number Example 5 Evaluate lim H2 x 2 3 Example 7 Evaluate lim x2 x XHO x x Now we ll look at limits as X gets big without bound approaches in nity 2x2 x2 Consider the function f x 1 What happens to x as we let the value of x get larger and larger x 10 50 100 1000 10000 100000 1000000 10000000 x We say that a function fx has the limit L as x increases without bound or as x approaches in nity written lim f x L if x can be made arbitrarily close to L by taking x large enough We say that a function fx has the limit L as x decreases without bound or as x approaches negative in nity written lim f x L if x can be made arbitrarily 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 8 Evaluate lim 2x 7 7 KAN x 7m 712 72 74 4 2 12 16 2 We can also nd limits at in nity algebraically or by recognizing the end behavior of a polynomial function Example 9 Evaluate lim 74x3 7 7x 5 Limits at in nity problems often involve rational expressions fractions Here s a technique we can use to evaluate limits at in nity 0 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 0 apply this theorem Theorem Suppose n gt 0 Then limi 0 and lim i 0 provided i is de ned xgtw x xgt7w x x After applying this limit we can determine what the answer should be YOUMUST KNOW THIS PROCEDURE 2 2 5 1 Example 10 Evaluate llmxz x Hm 3x 2x 7 Often students prefer to just lea1n some rules for finding limits at infinity 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 0 If the degree of the numerator is smaller than the degree of the denominator then lim fx 0 H gx 0 If the degree of the numerator is the same as the degree of the denominator then you can find lim by making a fraction from the leading coefficients of the 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 coefficients The notation lim f x 00 indicates that as the value of x increases the value of the Hm function increases without bound This limit does not exist but the co notation is more descriptive so we will use it 5 23 4 Example 11 Evaluate llme x Hm 4x 2x8 2 4 5 4 Example 12 Evaluate 11m xZ x HA x x1 Example 13 Evaluate 1im24x 5 Hm x 9x 9 Section 1057 Oneisided Limim and Continuity Sometimes We are only interested in the behavior of a mction When We look from one side and not from the other Consider the function fx Find thfx 6 x Now suppose We are only interested in nding a limit from one side of the target number For example We might look only at the values of x that are bigger than 0 In this case We are looking at a oneisided limit If We are looking at values of x that are bigger than 0 then We are considering a right hand limit Here s the notation for a onesided limit Where We are interesting only in values of x that are bigger than 0 We could also be interested in looking at the values of x that are smaller than zero If We are interested in looking only at the values of x that are smaller than 0 then We would be nding the le hand limit Here s the notation for a lefthand limit Our de nition of a limit from the last lesson is consistent with this information We say that lim f x L if and only if the function approaches the same value L from both the xgta left side and the right side of the target number This idea is formalized in this theorem Theorem Let f be a function that is defined for all values of x close to the target number a except perhaps at a itself Then lim fx L if and only if lim fx lim fx L Example 1 Consider this graph Find lim fx lim fx and ling fx if it eXists xaz Hr x We can also nd onesided limits from piecewise de ned functions x l x gt 3 Example 2 Suppose f x F1nd x2 5 x S 3 lim fx lim fx and lin21fx if it exists xgt239 xgt2 xgt 2x 3 x lt 1 x 2 x 2 139 Find lim fx lim fx and lim fx if it exists xarl xarf r1 Example 3 Suppose fx Continuity We will be interested in nding where a function is continuous and where it is discontinuous We ll look at continuity over the entire domain of the function over a given interval and at a speci c point Continuity at a Point Here s the general idea of continuity at a point a function is a continuous at a point if its graph has no gaps holes breaks or jumps at that point Stated a bit more formally A function f is said to be continuous at the point x a if the following three conditions are met 1 at is defined 2 lim fx exists 3 limfxfa You ll need to check each of these three conditions to determine if a function is continuous at a specific point 2x 3 x 2 l Example 4 Determme 1f f x 2 1s contmuous at x l x 4 x lt l If a function is not continuous at a point then we say it is discontinuous at that point We nd points of discontinuity by examining the function that we are given A function can have a removable discontinuity a jump discontinuity or an infinite discontinuity Example 5 Find any points of discontinuity State why the function is discontinuous at each point of discontinuity Continuity over an Interval A function is continuous over the interval a b if it is continuous at every point in the interval We ll state answers using interval notation 2 4 4 Example 6 Flnd the 1ntervals on wh1chf1s c0nt1nu0us f x x x Example 7 State where f is continuous using interval notation Tl 1 3 1 i Example 8 State where f is continuous using interval notation Sometimes we consider continuity over the entire domain of the function For many functions this is the entire set of real numbers Example 9 State where fx 3x4 5x2 2x 7is continuous

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