Calculus and Vectors – How to get an A+ 1.4 Limit of a Function A Left-Hand Limit If the values of y = f (x) can be made arbitrarily close to L by taking x sufficiently close to a with x < a , then: lim− f ( x) = L Ex 1. Use the function y = f (x) defined by the following graph to find each limit. x →a Read: The limit of the function f (x) as x approaches a from the left is L . Notes: 1. The function may be or not defined at a . 2. DNE stands for Does Not Exist. 3. L must be a number. 4. ∞ is not a number. a) lim − f ( x) = DNE x → −4 b) lim − f ( x) = 2 x →−2 c) lim − f ( x) = 2 x→−1 d) lim− f ( x) = 1 x→3 B Right-Hand Limit If the values of y = f (x) can be made arbitrarily close to L by taking x sufficiently close to a with x > a , then: lim+ f ( x) = R x →a Read: The limit of the function f ( x) as x approaches a from the right is R . Ex 2. Use the function y = f (x) defined at Ex 1. to find each limit. a) lim + f ( x) = 1 x→−1 b) lim+ f ( x) = ∞ ( DNE ) x→3 c) lim+ f ( x) = 3 x →1 d) lim + f ( x) = 2 x→−2 Notes: 1. R must be a number. ∞ is not a number. 2. The function may be or not defined at a . C Limit If the values of y = f (x) can be made arbitrarily close to l by taking x sufficiently close to a (from both sides), then: lim f ( x) = l Ex 3. Use the function y = f ( x) defined at Ex 1. to find each limit. a) lim f ( x) = DNE because lim − f ( x) = DNE Read: The limit of the function f ( x) as x approaches a is l . c) lim f ( x) = DNE because lim+ f ( x) = ∞ x →a x→−4 x→−4 b) lim f ( x) = DNE because lim − f ( x) = 2 ≠ lim + f ( x) = 1 x → −1 x →3 d) lim f ( x) = 1 x → −3 e) lim f ( x) = 2 x → −2 1.4 Limit of a Function ©2010 Iulia & Teodoru Gugoiu - Page 1 of 2 x →−1 x →3 x →−1 Calculus and Vectors – How to get an A+ Notes: f) lim f ( x) = 1 x→0 1. If lim+ f ( x) = lim− f ( x) then lim f ( x) does exist x →a x→ a x→a and L = R = l . 2. If lim+ f ( x) ≠ lim− f ( x) then lim f ( x) Does Not x →a x→a x→a Exist (DNE). 3. l must be a number. ∞ is not a number. 4. The function may be or not defined at a . Ex 4. Compute each limit. x2 12 1 = = a) lim− x→1 x + 1 1 + 1 2 12 1 x2 b) lim+ = = x →a x→1 x + 1 1 + 1 2 Notes: 2 2 1 1 x 1. In order to use substitution, the function must be c) lim = = x→1 x + 1 1 + 1 2 defined on both sides of the number a . D Substitution If the function is defined by a formula (algebraic expression) then the limit of the function at a point a may be determined by substitution: lim f ( x) = f (a ) 2. Substitution does not work if you get one of the following 7 indeterminate cases: 0 ∞ ∞ ∞ − ∞ 0×∞ 1 ∞ 0 00 0 ∞ d) lim− x − 2 = DNE x →2 e) lim+ x − 2 = 2 − 2 = 0 x →2 f) lim x − 2 = DNE x →2 E Piece-wise defined functions ⎧2 x − 3, x < 2 If the function changes formula at a then: ⎪ Ex 5. Consider f ( x) = ⎨0, x = 2 1. Use the appropriate formula to find first the left⎪ 2 side and the right-side limits. ⎩ x − 1, x > 2 2. Compare the left-side and the right-side limits to a) Find lim f ( x) . conclude about the limit of the function at a . x→2 Example: lim− f ( x) = lim− (2 x − 3) = (2)(2) − 3 = 1 x →2 x →2 ⎧ f1 ( x), x < a f ( x) = ⎨ lim f ( x) = lim+ ( x 2 − 1) = 2 2 − 1 = 3 x →2 + x →2 ⎩ f 2 ( x), x > a lim− f ( x) ≠ lim+ f ( x) ⇒ ∴ lim f ( x) = DNE x →2 x →2 x →2 L = f1 (a), R = f 2 (a) (if exist) b) Find lim f ( x) . x→0 lim f ( x) = lim (2 x − 3) = 2(0) − 3 = −3 x →0 x →0 ∴ lim f ( x) = −3 x →0 c) Draw a diagram to illustrate the situation. 5 y 4 3 2 1 x −5 −4 −3 −2 −1 1 −1 −2 −3 −4 Reading: Nelson Textbook, Pages 34-37 Homework: Nelson Textbook: Page 37 #4d, 5, 6, 7, 10cef, 11c, 15 1.4 Limit of a Function ©2010 Iulia & Teodoru Gugoiu - Page 2 of 2 2 3 4 5