Limit of a function and one - Sided limits

Limit of a Function and One-sided limits Mathematics 53 Institute of Mathematics (UP Diliman) Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 1 / 37 For today 1 Limit of a Function: An intuitive approach 2 Evaluating Limits 3 One-sided Limits Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 2 / 37 Introduction Given a function f (x) and a ∈ R, what is the value of f at x near a, but not equal to a? Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 4 / 37 Illustration 1 Consider f (x) = 3x− 1. What can we say about values of f (x) for values of x near 1 but not equal to 1? x f (x) 0 −1 0.5 0.5 0.9 1.7 0.99 1.97 0.99999 1.99997 x f (x) 2 5 1.5 3.5 1.1 2.3 1.001 2.003 1.00001 2.00003 Based on the table, as x gets closer and closer to 1, f (x) gets closer and closer to 2. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 5 / 37 Illustration 1 x f (x) 0 −1 0.5 0.5 0.9 1.7 0.99 1.97 0.99999 1.99997 −1 1 2 3 −1 1 2 3 4 Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 6 / 37 Illustration 1 x f (x) 2 5 1.5 3.5 1.1 2.3 1.001 2.003 1.00001 2.00003 −1 1 2 3 −1 1 2 3 4 As x gets closer and closer to 1, f (x) gets closer and closer to 2. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 7 / 37 Illustration 2 Consider: g(x) = 3x2 − 4x+ 1 x− 1 = (3x− 1)(x− 1) x− 1 = 3x− 1, x 6= 1 −1 1 2 3 −1 1 2 3 4 As x gets closer and closer to 1, g(x) gets closer and closer to 2. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 8 / 37 Illustration 3 Consider: h(x) = 3x− 1, x 6= 10, x = 1 −1 1 2 3 −1 1 2 3 4 As x gets closer and closer to 1, h(x) gets closer and closer to 2. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 9 / 37 Limit Intuitive Notion of a Limit a ∈ R, L ∈ R f (x): function defined on some open interval containing a, except possibly at a The limit of f (x) as x approaches a is L if the values of f (x) get closer and closer to L as x assumes values getting closer and closer to a but not reaching a. Notation: lim x→a f (x) = L Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 10 / 37 Examples f (x) = 3x− 1 −1 1 2 3 −1 1 2 3 4 lim x→1 (3x− 1) = 2 Note: In this case, lim x→1 f (x) = f (1). Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 11 / 37 Examples g(x) = 3x2 − 4x+ 1 x− 1 −1 1 2 3 −1 1 2 3 4 lim x→1 3x2 − 4x+ 1 x− 1 = 2 Note: Though g(1) is undefined, lim x→1 g(x) exists. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 12 / 37 Examples h(x) = 3x− 1, x 6= 10, x = 1 −1 1 2 3 −1 1 2 3 4 lim x→1 h(x) = 2 Note: h(1) 6= lim x→1 h(x). Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 13 / 37 Some Remarks Remark In finding lim x→a f (x): We only need to consider values of x very close to a but not exactly at a. Thus, lim x→a f (x) is NOT NECESSARILY the same as f (a). We let x approach a from BOTH SIDES of a. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 14 / 37 Some Remarks If f (x) does not approach any particular real number as x approaches a, then we say lim x→a f (x) does not exist (dne). e.g. H(x) =  1, x ≥ 0 0, x < 0 −3 −2 −1 1 2 3 1 2 3 0 lim x→0 H(x) = 0? No. lim x→0 H(x) = 1? No. lim x→0 H(x) dne Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 37 Limit Theorems Theorem If lim x→a f (x) exists, then it is unique. If c ∈ R, then lim x→a c = c. lim x→a x = a Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 37 Limit Theorems Theorem Suppose lim x→a f (x) = L1 and limx→a g(x) = L2. Let c ∈ R, n ∈ N. lim x→a[ f (x)± g(x)] = limx→a f (x)± limx→a g(x) = L1 ± L2 lim x→a[ f (x)g(x)] = ( lim x→a f (x) ) ( lim x→a g(x) ) = L1L2 lim x→a[c f (x)] = c limx→a f (x) = cL1 lim x→a f (x) g(x) = lim x→a f (x) lim x→a g(x) = L1 L2 , provided L2 6= 0 lim x→a ( f (x)) n = ( lim x→a f (x) )n = (L1)n Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 37 Evaluate: lim x→−1 (2x2 + 3x− 4) lim x→−1 (2x2 + 3x− 4) = lim x→−1 2x2 + lim x→−1 3x− lim x→−1 4 = 2 ( lim x→−1 x2 ) + 3 ( lim x→−1 x ) − lim x→−1 4 = 2 ( lim x→−1 x )2 + 3 ( lim x→−1 x ) − lim x→−1 4 = 2(−1)2 + 3(−1)− 4 = −5 In general: Remark If f is a polynomial function, then lim x→a f (x) = f (a). Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 37 Evaluate: lim x→−2 4x3 + 3x2 − x+ 1 x2 + 2 lim x→−2 4x3 + 3x2 − x+ 1 x2 + 2 = lim x→−2 (4x3 + 3x2 − x+ 1) lim x→−2 (x2 + 2) = 4(−2)3 + 3(−2)2 − (−2) + 1 (−2)2 + 2 = −17 6 Remark If f is a rational function and f (a) is defined, then lim x→a f (x) = f (a). Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 37 Theorem Suppose lim x→a f (x) exists and n ∈ N. Then, lim x→a n √ f (x) = n √ lim x→a f (x), provided lim x→a f (x) > 0 when n is even. lim x→3 √ 3x− 1 = √ lim x→3 (3x− 1) = √ 8 = 2 √ 2 lim x→−1 3 √ x+ 4 x− 2 = 3 √−1+ 4 −1− 2 = −1 lim x→7/2 4 √ 3− 2x dne lim x→2 √ x2 − 4 =? (for now) Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 37 Evaluate: lim x→3 ( 2x2 −√5x+ 1 x3 − x+ 4 )3 lim x→3 ( 2x2 −√5x+ 1 x3 − x+ 4 )3 =  limx→3 2x2 − √ lim x→3 (5x+ 1) lim x→3 (x3 − x+ 4)  3 = ( 18− 4 28 )3 = 1 8 Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 37 Consider: g(x) = 3x2 − 4x+ 1 x− 1 . From earlier, limx→1 g(x) = 2. Can we arrive at this conclusion computationally? Note that lim x→1 ( 3x2 − 4x+ 1 ) = 0 and lim x→1 (x− 1) = 0. But when x 6= 1, 3x 2 − 4x+ 1 x− 1 = (3x− 1)(x− 1) x− 1 = 3x− 1. Since we are just taking the limit as x → 1, lim x→1 3x2 − 4x+ 1 x− 1 = limx→1(3x− 1) = 2. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 37 Definition If lim x→a f (x) = 0 and limx→a g(x) = 0 then lim x→a f (x) g(x) is called an indeterminate form of type 0 0 . Remarks: 1 If f (a) = 0 and g(a) = 0, then f (a) g(a) is undefined! 2 The limit above MAY or MAY NOT exist. 3 Some techniques used in evaluating such limits are: Factoring Rationalization Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 37 Examples Evaluate: lim x→−2 x3 + 8 x2 − 4 ( 0 0 ) lim x→−2 x3 + 8 x2 − 4 = limx→−2 (x+ 2)(x2 − 2x+ 4) (x+ 2)(x− 2) = lim x→−2 x2 − 2x+ 4 x− 2 = 4+ 4+ 4 −2− 2 = −3 Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 37 Examples Evaluate: lim x→4 x2 − 16 2−√x ( 0 0 ) lim x→4 x2 − 16 2−√x = limx→4 x2 − 16 2−√x · 2+ √ x 2+ √ x = lim x→4 (x2 − 16)(2+√x) 4− x = lim x→4 (x− 4)(x+ 4)(2+√x) 4− x = lim x→4 [−(x+ 4)(2+√x)] = (−8)(4) = −32 Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 37 Examples Evaluate: lim x→−1 3 √ x+ 1 2x+ 2 ( 0 0 ) lim x→−1 3 √ x+ 1 2x+ 1 = lim x→−1 3 √ x+ 1 2x+ 1 · 3√x2 − 3√x+ 1 3√x2 − 3√x+ 1 = lim x→−1 x+ 1 2(x+ 1)( 3 √ x2 − 3√x+ 1) = lim x→−1 1 2( 3 √ x2 − 3√x+ 1) = 1 6 Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 37 Illustration 4 Consider: f (x) =  3− 5x2, x < 1 4x− 3, x ≥ 1 As x → 1, the value of f (x) depends on whether x 1. −4 −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 4 0 As x approaches 1 through values less than 1, f (x) approaches −2. As x approaches 1 through values greater than 1, f (x) approaches 1. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 29 / 37 Illustration 5 Consider: g(x) = √ x −2 −1 1 2 3 −1 1 2 0 ( )( )( )( ) Since there is no open interval I containing 0 such that g(x) is defined on I, we cannot let x approach 0 from both sides. But we can say something about the values of g(x) as x gets closer and closer to 0 from the right of 0. Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 37 One-sided Limits Intuitive Definition The limit of f (x) as x approaches a from the left is L if the values of f (x) get closer and closer to L as the values of x get closer and closer to a, but are less than a. Notation: lim x→a− f (x) = L Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 37 One-sided Limits Intuitive Definition The limit of f (x) as x approaches a from the right is L if the values of f (x) get closer and closer to L as the values of x get closer and closer to a, but are greater than a Notation: lim x→a+ f (x) = L Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 32 / 37 Theorem lim x→a f (x) = L if and only if limx→a− f (x) = lim x→a+ f (x) = L Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 33 / 37 f (x) =  3− 5x2, x < 1 4x− 3, x ≥ 1 −4 −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 4 0 lim x→1− f (x) = lim x→1− (3− 5x2) = −2 lim x→1+ f (x) = lim x→1+ (4x− 3) = 1 lim x→1 f (x) dne Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 34 / 37 Examples g(x) = √ x −2 −1 1 2 3 −1 1 2 0 Based on the graph, lim x→0+ √ x = 0 lim x→0− √ x dne lim x→0 √ x dne Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 35 / 37 Examples Let p(x) =  √ 5− 2x, x ≤ 2 x2 − 2x 2x− 4 , x > 2 lim x→2− p(x) = lim x→2− √ 5− 2x = √ 1 = 1 lim x→2+ p(x) = lim x→2+ x2 − 2x 2x− 4 = limx→2+ x(x− 2) 2(x− 2) = limx→2+ x 2 = 1 lim x→2 p(x) = 1 lim x→3 p(x) = lim x→3 x2 − 2x 2x− 4 = 9− 6 6− 4 = 3 2 Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 36 / 37 Exercises 1 Evaluate: lim x→4 3x− 12 3−√2x+ 1 2 Find lim x→−1 f (x) given: f (x) =  x2 + 1 x− 1 , if x < −1 1−√x+ 5, if x ≥ −1 3 Evaluate: lim x→2/3 ( 6x− 4 3x2 + 4x− 4 + 1 3x+ 2 ) Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 37 / 37