Limits and Continuity of a Function (Module 1) Mat051 AY 2020-2021 (1st semester) This material is not for sale and is owned by the Department of Mathematics and Statistics, College of Science and Mathematics of Mindanao State University-Iligan Institute of Technology. LIMITS AND CONTINUITY OF FUNCTIONS The limit concept is considered to be the most essential concept in calculus. It is used to define continuity, derivatives and integrals. Consider the function defined by . When , an indeterminate. Let’s look at how the values of the function behaves when gets closer and closer to . 1 3 ⺁ 2.5 4.5 1.5 3.5 2.4 4.4 1.6 3.6 2.3 4.3 1.7 3.7 2.2 4.2 1.8 3.8 2.1 4.1 1.9 3.9 2.01 4.01 1.99 3.99 2.001 4.001 1.999 3.999 Here, we can see that as function ⺁ 2.0001 4.0001 gets closer and closer to , the value of the ⺁ gets closer and closer to 4. A “hole” or a skip Figure 1 Graph of the function Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS If we say that is the limit of that the value of ⺁ as approaches the number ⺁ can be made close to by taking words, we can make the absolute difference of very small by making the absolute difference of close to . In other ⺁ and , written and , it means or , very small. This is formally stated in the following definition. Definition Let be a function which is defined at all containing , except possibly at itself. The limit of is , written lim if for every , however small, there exists a whenever on the open interval ⺁ as approaches such that . The following example uses the definition to prove that the given function has the indicated limit. Example Solution: Prove that the lim t. We need to show that for every there exists a that Now, t t whenever . Hence, we must show that whenever or equivalently, Choose whenever . . Then we have whenever or, equivalently, This proves that lim t whenever t. Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS such Limit Theorems 1. If lim and lim , then . ᔠ ᔠ . 2. If and are constants, then lim 3. If is a constant, then for any real number , lim 4. For any real number , lim 5. If lim and lim lim 6. If lim and lim . , then ᔠ ⺁ , then ᔠ lim ᔠ lim f1 ( x) f 2 ( x) lim f1 ( x) lim f 2 ( x) L1 L2 xa 7. If lim and 8. If lim , then provided n xa lim Moreover, lim 1. lim lim , , ⺁ , then lim lim exists, then lim lim have ⺁ lim are two functions such that , and if lim Example lim lim and lim and xa is any positive integer, then we L R . 9. If lim 10. If lim ⺁ for all exists. . Evaluate the following limits. ᔠ ᔠ ᔠ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS Solution 2. lim t ᔠ ᔠ lim ᔠ ᔠ ⺁ ᔠ ⺁ᔠ ⺁ ᔠ Solution lim t t ᔠ ᔠ is undefined, so apply Theorem 10 above, lim t ᔠ lim t t ᔠ t t ᔠ lim t lim ᔠ t ᔠ t t 3. lim ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ t t t ᔠ t ᔠ t ᔠ ᔠ t ᔠ ᔠ t ᔠ ᔠ ᔠ ᔠ ᔠ Solution lim ᔠ lim lim lim lim ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS lim lim tᔠ 4. lim ᔠ ᔠ ᔠ ᔠ ᔠ Solution lim t ᔠ ᔠ lim lim ᔠ ⺁ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ⺁ ⺁ᔠ Exercises 1. lim t t 2. lim 3. lim ᔠ 4. lim 5. lim 7. lim 9. 10. lim ᔠ t t lim lim tᔠ ᔠ t ᔠ t 6. lim 8. t tᔠ t ᔠ ᔠ t t t t t Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS ONE-SIDED LIMITS Consider the function if if ᔠ Its graph is shown below approaches approaches approaches Figure 2 Graph of the function Notice that as approaches that approaches from the right, approaches approaches ᔠ if if from the left, ⺁ gets closer to ⺁ gets closer to . In symbols, through values greater than through values less than and but when ᔠ means means that . We shall now define one-sided limits formally. Definition (Right-Hand Limit) Let be a function which is defined for every ⺁. Then the limit of if for every ⺁ as approaches limᔠ , however small, there exists a whenever in some open interval form the right is , written such that . Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS Definition (Left-Hand Limit) Let be a function which is defined for every ⺁. Then the limit of if for every ⺁ as approaches form the left is , written lim , however small, there exists a whenever in some open interval such that . Remark The limit theorems discussed earlier still hold if “ ᔠ by “ ” or “ ” is replaced ”. The following theorem gives a relationship between the ordinary limit and the one-sided limits. Theorem lim exists if and only if limᔠ and lim Moreover, lim Example limᔠ Solution Since approaches 0 from the right, it takes on positive values, that is, , so limᔠ Solution lim Evaluate the following limits. 1. limᔠ 2. lim both exist and are equal. limᔠ ᔠ lim ᔠ lim lim ⺁ ⺁ ⺁ ⺁ t t ᔠ ⺁ ⺁ ᔠ ⺁ ⺁ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS lim t 3. ᔠ lim ᔠ Solution Note that t ᔠ ⺁ ⺁ ᔠ ⺁ ⺁ approaches -1 from the left. That is, on negative values, so ᔠ ⺁ and ᔠ lim ᔠ 4. Given determine if lim ᔠ ᔠ ᔠ takes ᔠ ᔠ lim , t exists. Solution We need to find the limit of from the left and from the right and see if they are equal. Now, limᔠ lim limᔠ lim ᔠ t ᔠ t t Since the left-hand side and right-hand side limits are not the same, the does not exist. lim Exercises A. Evaluate the following limits. 1. 2. 3. lim lim t t t lim t tᔠ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS 4. 5. 6. 7. ᔠ lim ᔠ limᔠ t lim t lim t t ⺁ B. In problems 1 to 3, find (a) limᔠ 1. determine if lim exists. C. Given find the following 2. 3. . In each case, t 2. 1. and (b) lim lim ᔠ 4. 5. lim lim if it exists⺁ lim ᔠ ᔠ lim 6. lim , if it exists⺁ INFINITE LIMITS Consider the function defined by gets closer and closer to . Its graph below shows that as , the value of the function bigger, it increases without bound. We use ᔠ lim ᔠ ⺁ gets bigger and to denote this increase, that is, . Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS Figure 3 Graph of In the same way, if the value of the function approaches , then we say that the limit is ⺁ decreases without bound as . The following are the formal definitions of infinite limits. Definition ᔠ Let Limit⺁ be a function which is defined at all containing , except possibly at bound as approaches , written if for every positive number whenever itself. We say that lim , there exists a ⺁ increases without such that . Definition Let Limit⺁ be a function which is defined at all containing , except possibly at bound as approaches , written if for every positive number whenever ⺁ ᔠ on the open interval lim itself. We say that ⺁ , there exists a on the open interval ⺁ decreases without such that . Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS Theorem If is a positive integer, then limᔠ ᔠ lim if is odd if is even ᔠ Theorem If , and if lim to zero, then (1) If and if (2) If and if (3) If and if (4) If and if ⺁ and lim , where ⺁ is a constant not equal through positive values of ⺁ lim ᔠ ⺁, then through positive values of ⺁ lim ⺁, then through negative values of ⺁ lim through negative values of ⺁ lim ᔠ ⺁, then ⺁, then Theorem If lim ⺁ ᔠ and lim ᔠ and lim ⺁ , where lim ᔠ is a real number, then ⺁ ᔠ Theorem If lim (1) If (2) If ⺁ , then lim , then lim ⺁ , where ⺁ ⺁ is a nonzero real number, then ᔠ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS Theorem If lim (1) If ⺁ (2) If and lim , then lim , then lim ⺁ , where ⺁ ⺁ is a nonzero real number, then ᔠ Example Evaluate the following limits. 1. limᔠ Solution positive limᔠ limᔠ approaches 0 through positive values of ᔠ 2. limᔠ Solution Notice that limᔠ and limᔠ resulting to we’ll find an equivalent function as follows, limᔠ limᔠ limᔠ limᔠ limᔠ ᔠ ᔠ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS . So ᔠ Now, as , through positive values, that is limᔠ 3. limᔠ Solution limᔠ ᔠ Notice that limᔠ ᔠ and ᔠ . But there’s no way we can subtract (or add) both infinite values. So, transforming the function, limᔠ ᔠ Now, as , limᔠ limᔠ through positive values. This results to limᔠ ᔠ 4. limᔠ Solution From the first factor of the function, notice that limᔠ and from the second factor limᔠ limᔠ ᔠ . Thus, ᔠ Exercises 1. lim t 2. limᔠ 3. lim ᔠt t ᔠ t ᔠ t ᔠ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS ᔠ 4. limᔠ t t 5. lim 6. lim lim 10. 11. ᔠ ᔠ t ᔠt ᔠ t lim ᔠ 12. lim t ᔠ t⺁ ᔠ ⺁ lim lim ᔠ t t 8. limᔠ 9. t t t 7. ᔠ ᔠ t ᔠ t LIMITS AT INFINITY Definition Limit at ᔠ Let interval ᔠ be a function which is defined at every number in some open ⺁. The limit of if for any lim ᔠ increases without bound is , ⺁ such that . Definition Limit at Let ⺁ , ⺁ be a function which is defined at every number in some open ⺁. The limit of if for any whenever ⺁ as , there exists a number whenever interval ⺁ ⺁ as lim decreases without bound is , ⺁ , there exists a number such that . Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS ⺁ , Theorem If is a positive integer, then lim ᔠ lim Example Evaluate the following limits. 1. lim t Solution: ᔠ ᔠ lim t ᔠ ᔠ lim lim lim 2. t lim t ᔠ t t ᔠ ᔠ ᔠ ᔠ ᔠ Solution: lim Note that lim since . Thus, Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS lim lim lim lim lim 3. lim ᔠ Solution: lim ᔠ ᔠ ᔠ ᔠ ᔠ lim ᔠ ᔠ ᔠ lim ᔠ lim ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ lim ᔠ lim ᔠ ᔠ ᔠ lim ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS Exercises 1. lim 2. 3. ᔠ lim 6. 7. 8. 9. t tᔠ ᔠ ᔠ ᔠ ᔠ lim ᔠ 4. lim 5. tᔠ ᔠ lim lim ᔠ lim ᔠ t t tᔠ t ᔠ ᔠ lim ᔠ 10. lim ᔠ 11. lim ⺁ t ᔠ ᔠ lim ᔠ ᔠ tᔠ t t ᔠ ᔠ ᔠ ᔠ ᔠt⺁ ᔠ ᔠ ᔠ ⺁ ᔠ t 12. lim ᔠ ⺁ Continuity of a Function Definition (Continuity at ) The function is said to be continuous at the number if the following three conditions are satisfied 1. 2. lim 3. lim exists; exists ⺁ Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS If one or two of these conditions fail to hold, then we say that the function discontinuous. If either lim is discontinuous at a number ⺁ or but lim is exists, then ⺁ does not exist. If this happens, then we say that the discontinuity is removable. If the discontinuity at is not removable, then it is called an essential discontinuity. Theorem If and and are two functions which are continuous at are continuous at . If , then ᔠ , then is also continuous at . ⺁ Example Determine if the given function is continuous at the indicated number . If it is discontinuous at , determine if the discontinuity is removable or essential. if 1. Solution: ; if ⺁ does not exist since lim limᔠ lim limᔠ ⺁ lim Therefore, the function limᔠ lim is discontinuous and the discontinuity is essential. 2. Solution: if if ; Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS lim ⺁ lim ⺁ Therefore, t 3. ᔠ lim lim lim ᔠt ᔠt is continuous at ; Solution: Examining each condition for continuity, we have is not defined lim lim t t t ᔠ ᔠ t ᔠ ᔠ t ᔠ t t t t ᔠ lim Therefore, the function lim at t ᔠ t ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ ᔠ t ᔠ ᔠ ᔠ t lim ᔠ t ᔠ ᔠ t ᔠ t ᔠ ᔠ t ᔠ ᔠ ᔠ ᔠ t ᔠ is not continuous continuos at ᔠ ⺁ t . However, since the exists, the discontinuity is removable. To make it continuous , the function can be redefined as, Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS t ᔠ t if if Exercises Determine if the given function is continuous at the indicated number . If it is discontinuous at , determine if the discontinuity is removable or essential. ᔠ 1. at ; and at ᔠ 2. at t and at 3. 4. ᔠt⺁ 5. ᔠ t ᔠt 7. ᔠ ⺁ ; at ; at t if if ᔠ 9. at ᔠ 10. t ; at if ᔠ 8. t ; at if 6. 11. t t t; t ; at if if ; at if at ᔠ at Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS 12. 13. Let at ᔠ be defined by ᔠ For what value of is 14. For what values of continuous at ᔠ if if and a continuous function at and t and ? is , defined by if t ᔠ if t if ? Intermediate Value Theorem Theorem (Intermediate Value Theorem) Suppose that between is continuous on the closed interval ⺁ and ⺁. Then there is a number e and is any number such that . The intermediate value theorem says that a function must take on every value between its endpoints at least once provided the function is continuous on a compact interval. It means that a continuous function over any values between to jump across the line ⺁ and on ⺁. Otherwise, the graph of e cannot skip would need , something that continuous functions cannot do. A function may take on a given value more than once. Although these geometric representations make the IVT seem reasonable, the proof is more complicated than one might imagine and an interested reader may refer to an advanced calculus or real analysis text. The following result is a special case of the Intermediate Value Theorem. It is also the basis of a Root-Finding algorithm called the Bisection Method. Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS Corollary Suppose that is continuous on ⺁ such that e and . . Then there exists Example Use the Intermediate Value Theorem to verify that the following functions have zero/s in the given interval. Then use the method of bisections to find an interval of length t that contains the zero. 1. t 2. 3. 4. t t cos ᔠ Squeeze Theorem and Limits Involving Sine and Cosine Example 1. Show that lim sin . Solution: It can be seen that t sin Since sin , for all sin sin , we have t sin , for all and . , for all By Squeeze Theorem, lim lim sin lim lim sin Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS 2. Show that lim cos Solution: For all . , cos Thus, cos cos , for all cos , for all and . Let and Hence, for cos Now, lim By Squeeze Theorem, lim lim cos Example Evaluate the following limits. 1. lim sin Solution lim sin ᔠ ᔠ This limit is of the form . lim sin lim ᔠ sin lim lim sin sin lim lim ᔠ ᔠ ᔠ where note Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS 2. lim t t cos Solution: lim cos cos ᔠ cos lim lim lim 3. lim cos lim sin ᔠ cos sin lim ᔠ tan ᔠ cos ᔠ cos ᔠ cos Solution: lim tan lim lim sin cos sin cos lim cos Exercises Evaluate the following limits. 1. lim 2. sin lim sin t ᔠ ᔠ ⺁ 3. lim t cot t 4. lim tan Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS cos 5. lim t t 6. lim csc ⺁ 7. lim cot 9. lim cos 11. lim sin cos sin cos sin 8. lim t tant 10. lim t sin 12. lim t 13. lim ttan 14. lim sin Department of Mathematics and Statistics, CSM, MSU-IIT Prepared by MAT051 Faculty: LMB, ECC, MBF,BBM, FRS
