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Calculus Limits Lesson 2 Bell Activity A. Use your calculator graph to find: 1. lim x2 (3x 2) 2. lim x3 (2 x 5) x 3. lim x4 (4 ) 2 2 ( x 4) 4. lim x 2 ( x 2) B. Without Calculator, Find 1. f(2) if f(x) = 3x – 2 2. f(-3) if f(x) = 2x + 5 3. f(4) if f(x) = 4 – x/2 4. 2 ( x 4) f(2) if f(x) = ( x 2) Notice: In #4, both the numerator and denominator are 0. This is called an indeterminate form and thus cannot be evaluated. The limit may be estimated or found by creating a table of values very close to the approaching x value. ( x 3x 2) lim =1 ( x 2) x 2 2 X 1.75 1.9 1.99 Y .75 .9 .99 2 2.001 1.001 2.01 1.01 2.1 1.1 *** lim x3 (2 x 5) =1 X 2.8 2.9 2.99 3 3.001 3.01 3.1 Y .6 .8 .98 1 1.002 1.02 1.2 It would seem from this example that at times a limit of a function as x approaches a particular # can simply be obtained by finding f(that #). What do you think allows this function to have the limit as x approaches 3 and f(3) to be the same whereas the first problem does not? lim x3 (2 x 5) x 3x 2 x2 =1 2 lim x2 =1 Which of these can be substituted directly to find the limit? 1. lim x 3 ( x 2 2) =11 2 x 1 2. lim =3 x2 x 1 2 x 1 3. lim x 1 =2 x 1 Even though # 3’s limit can’t be found by substituting directly we can still get the limit from a graph or a table. Let’s go back to a problem where the substitution would not work to find the limit. Perhaps we could modify the problem before we substituted. 2 lim x1 x 1 x 1 ( x 1)( x 1) lim x 1 x 1 lim x1 ( x 1) =2 Modify these and find the limit by substituting: x x6 2 x 9 lim x3 x 1 x 1 3 2 lim x1 Let’s verify these with the calculator graph. Find the limit: x 1 x 1 2 lim x1 Let’s verify with the calculator graph. Find the limit: Let’s verify with the calculator graph. There are other times when we cannot find the limit by either substituting or factoring. lim x 0 In this case, Rationalize the Numerator x 1 1 x And, of course, there are problems which merely need to be simplified before substituting. ( x x) x x 3 lim x0 3 Find the limit: Let’s verify with the calculator graph. Find the limit: Let’s verify with the calculator graph. Find the limit: Let’s verify with the calculator graph. Properties of Limits If L, M , c, and k are real numbers and lim f x L x c 1. Sum Rule : and lim g x M , then x c lim f x g x L M x c The limit of the sum of two functions is the sum of their limits. 2. DifferenceRule : lim f x g x L M x c The limit of the difference of two functions is the difference of their limits. Slide 2- 17 Properties of Limits continued 3. Product Rule: lim f x g x L M x c The limit of the product of two functions is the product of their limits. 4. Constant Multiple Rule: lim k f x k L x c The limit of a constant times a function is the constant times the limit of the function. 5. Quotient Rule : lim x c f x L , M 0 g x M The limit of the quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. Slide 2- 18 Properties of Limits continued 6. If r and s are integers, s 0, then Power Rule : r s r s lim f x L x c r s provided that L is a real number. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number. Other properties of limits: lim k k x c lim x c x c Example Properties of Limits Use any of the properties of limits to find lim 3x3 2 x 9 x c lim 3x3 2 x 9 lim3x3 lim 2 x lim9 x c x c x c 3c3 2c 9 x c sum and difference rules product and multiple rules Assignment Text p. 67, # 1 – 43 odds