Calc for BusLife Sciences
Calc for BusLife Sciences MATH 1314
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This 7 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1314 at University of Houston taught by Mary Flagg in Fall. Since its upload, it has received 19 views. For similar materials see /class/208403/math-1314-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
Math 1314 Lesson 2 OneSided Limits and Continuity OneSided Limits Sometimes we are only interested in the behavior of a function when we look from one side and not from the other Example 1 Consider the function fx Find linngc x H Now suppose we are only interested in looking at the values of x that are bigger than 0 In this case we are looking at a onesided limit We write lim f x This is called a righthand limit because we are looking at values xgt on the right side of the target number In this case 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 The values of x that are smaller than 0 are to the le of 0 on the number line hence the name We write lim f x xgt 7 Math 1314 Lesson 2 Page 1 of7 In this case 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 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 de ned for all values of x close to the target number 1 except perhaps at 11 itself Then lim f x L if and only if lim f x lim f x L Example 2 Consider this graph Find lirgr fx lirgr fx and lint fx if it exists We can also find onesided limits from piecewise de ned functions Math 1314 Lesson 2 Page 2 of7 x 3 gt 2 Example 3 Suppose fx 2 Find lim fx lim fx and lim fx if x 5 2 xgt239 xgt2 xgt2 it exists 2x 3 x lt 1 x 2 x Z 139 Find lim fx lim fx and lini1 fx if it exists rarl xgt71 x Example 4 Suppose fx Math 1314 Lesson 2 Page 3 of7 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 liffl xfa LA You ll need to check each of these three conditions to determine if a function is continuous at a specific point 2x 3 x Z l Example 6 Determme 1f f x 2 1s contmuous at x l x 4 x lt 1 Math 1314 Lesson 2 Page 4 0f7 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 in nite discontinuity Example 7 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 Z 7 Example 8 Find the intervals on which f is continuous f x W x i x Math 1314 Lesson 2 Page 5 of7 Example 9 State where f is continuous using interval notation Example 10 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 Math 1314 Lesson 2 Page 6 of 7 Example 11 State where fx 3x4 5x2 2x 7is continuous From this lesson you should be able to Say what we mean by a onesided limits Find a onesided limit from the graph of a function Find a onesided limit from a piecewisedefined function Find a onesided limit from a function Determine if a function is continuous at a point Find points of discontinuity over an interval or over the domain of the function given either a graph of the function or the function itself State intervals where a function is continuous given either a graph of the function or the function itself Math 1314 Lesson 2 Page 7 of7
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