The general limit notation is lim f ( x) as x approaches c is L. , which is read the limit of f(x) MA Lesson 28 Notes Summer 2016

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1 MA Lesson 8 Notes Summer 016 In everyday language, people refer to a speed it, a wrestler s weight it, the it on one s endurance, or stretching a spring to its it. These phrases all suggest that a it is a bound, which on some occasions may not be reached but on other occasions may be reached or eceeded. Consider a spring that will break or completely stretch out only if a weight of 10 pounds or more is attached. To determine how far the spring will stretch without breaking, you could attach increasingly heavier weights and measure the spring length s for each weight w, as shown in the picture below. If the spring length approaches a value of L, then it is said that the it of s as w approaches 10 is L. A mathematical it is much like the it of the spring described. As a weight hanging from a spring approaches 10 pounds, the length of the stretched spring will approach a certain number called the it. Let us suppose that it is 8 inches. (If any more weight than 10 pounds is put on the spring, it will break or stop stretching beyond 8 inches.) We could say the it of the length of the stretched spring as the weight approaches 10 pounds is 8 inches. This would be written as (spring length) 8 w10 (the it of the length of the spring as the weight w approaches 10 pounds is 8 inches). The general it notation is f ( ) as approaches c is L. c L, which is read the it of f() 1

2 Finding Limits: There are many different strategies used to find its. One approach is to evaluate the function for numbers very close to c, slightly greater than c (called approaching from the right ) and/or slightly less than c (called approaching from the left ). Sometimes a table is used, such as in the eample below. E 1: Find f ( ) where f ( ) 3 or ( 3) f () Approaching from the left Approaching from the right You can easily see from the table that the closer the value is to 4, the function value ( + 3) is approaching 11. We say ( 3) As seen above, the first strategy for finding a it is using a table such as above and approaching the value from the left and from the right. Coincidently, a direct substitution of 4 into the function epression + 3 also yields a value of 11. ( 3) (4) A second strategy for finding a it (works with many polynomial, rational, radical epressions) is direct substitution. Note: When using direct substitution, the it value is the function value or the y- value. 9 9 E : Find g( ) where g( ) or g() Approaching from the left Approaching from the right = Notice that direct substitution would not work in eample above. Direct substitution yields 0. The epression 0 is called an indeterminant form. Whenever an indeterminant 0 0 form results, the it may or may not eist. Another strategy must be tried.

3 One-sided Limits: In general for a it to eist, the same value must be approached from the left and from the right. However, at times, mathematicians and statisticians are only interested in approaching from the left or approaching from the right. These are called one-sided its, a left-sided it or a right-sided it. Below is the notation for these types of its. f( ) left-sided it notation c c f( ) right-sided it notation E 3: Use the table below to determine (if possible),, and (approaching from the left) (approaching from the right). = = = 3 E 4: Use the table below to determine (if possible) = E 5: Use the table below to determine (if possible) undefined 3 4 = Important Conclusions: 3

4 1) Saying that the it of f() or an epression as approaches c is L means that the function or epression value gets very, very close to L as gets very, very close to c. ) For a it L to eist, you must allow to approach c from either the left side of c or the right side of c and these two values must be the same. 3) The function or epression does not have to be defined at c in order fo the it as c to eist. In other words, a it may eist even though the function value does not eist. Techniques for Finding Limits: A With many functions or epressions, a table similar to those used in previous eamples may be used. Approach from the left and from the right. If both are the same value, that is the it. A table may also be used to find one-sided its; its only approaching from the left or only approaching from the right. B With many functions or epressions, direct substitution may be used. Below are a few eamples. ( ) ( ) ( ) 4 6 C (n 4) ( 3 4) 4 n3 a a5 n 5 () or n n If direct substitution yields the inderterminant form 0, you can sometimes write an 0 equivalent epression by simplifying and then try direct substitution. Below are a few eamples Try simplifying the rational epression first ( 3)( 3) ( 3) c 4c 5 5 4(5) 5 0 Try simplifying the rational epression first. c5 c c 4c 5 ( c 5)( c 1) ( c 1) c5 c5 c5 c5 c5 3( ) (3 ) 3( 0) (3 ) 0 Try simplifying first ( ) (3 ) E 6: Find each it, if possible. If the it does not eist, write DNE. 4

5 a) ( 3) b) 5 (a 5a 7) a3 c) 1 d) 30 m 1 1 m m m e) f) 4 6 g) h)

6 i) t t 3t10 t 4 j) 4( c) 3 (4 3) c0 c k) ( r r) ( r r) 1 ( r r 1) r 0 r E 7: Find the following one-sided its. 6 a) ( 1) 9 b) (sin 3) 0 6

7 7c c) c0 c 5c d) 4 ( 4) e) 5 0 7