Warm-Up: Sketch a graph of the function and use the table to find the lim 3x 2. lim 3x. x 5. Level 3 Finding Limits Algebraically.

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1 Warm-Up: 1. Sketch a graph of the function and use the table to find the lim 3x 2 x 5 lim 3x 2. 2 x 1

2 finding limits algibraically Ch.15.1 Level 3 2

3 Target Agenda Purpose Evaluation TSWBAT: Find the limit of a function given Limit Laws. Find the limit using Factoring, and Simplifying Warm-Up Lesson BAT: Developing our Life Skill Get ready for calculus. Extend knowledge functions, make connections, apply knowledge to new situations

4 This is CRAZIER! By HAND!!!! What is the lim 3x 2 4x + 2? x 5 What is the limit x 4 + x 2 2? x 1 4 4x BIGGER 4

5 Finding the Limit Algebraically Basic yet Necessary Rules lim C = C x a lim x = a x a lim x k = a k x a k k lim x = a x a 5

6 THE Limit Laws Let c be a constant lim f(x) = M and x a lim g(x) = L x a lim [f(x) + g(x)] = M + L x a lim [f(x) g(x)] = M L x a lim c f(x) = c M x a lim [f(x) g(x)] = M L x a lim x a f(x) g(x) = M L lim g(x) lim f(x) k = M k x a k k lim f(x) = M x a k > 0 a,k > 0 6

7 Basically: Whatever you do to the function or functions, you can do directly to their limits 7

8 Finding the Limit Algebraically Which rule did they use? Match them! lim C = C x a lim x = a x a lim x k = a k x a k k lim x = a x a ex. lim (x 4 ) = x 16 ex. lim x = x 16 ex. lim x = x 16 ex. lim 5 = x 16 8

9 Combining Limit Laws Example lim 5(t + 3) 8 x 4 9

10 Summarizing TIME! 10

11 Finding the Limit Algebraically Substitute a in for x, that's the limit! lim f(x) = f(a) x a a Domain of f ex. lim 5(t + 3) 8 x 4 ex. lim (3x 2 4x + 2) x 6 **If this is true for all points in the function, then it is continuous! 11

12 Canceling Common Factors Use this tool when dividing by 0 Because: Law 5 Quotient and direct substitution cant be used to find the limits because we would be dividing by 0 ex. lim x 2 x 2 x

13 Finding the Limit by Simplifying Use when the numerator is not simplified or if you can cancel out the denominator completely Because: Law 5 Quotient and direct substitution cant be used to find the limits because we would be dividing by 0 ex. lim (6+2x)2 36 x 0 2x 13

14 Finding the Limit by Finding: Left and Right hand limits separately Use when there is a piecewise function or absolute values in the function Because:{3x if x < 0 x 0+ p(x) = x 0 x 0 x 4 x 4 x 4+ x 3 x 3 if 0 < x < 4 6 x if x > 4 14

15 Finding the Limit by Finding: Left and Right hand limits separately Use when there is a piecewise function or absolute values in the function 4 if x=2 Because:{ p(x) = 1/3x if x 2 p(2) x 2+ x 2 x 2 15

16 Try it again! Left and Right hand limits separately Use when there is a piecewise function or absolute values in the function Because:{1 if x < 2 p(x) = p(2) 3 if x=2 1 if x > 2 x 2+ x 2 x 2 16

17 Evaluation: f(x) = (x 2 +5x + 6)(x+4) x 2 +6x+8 1. lim f(x) = x 4 Practice! pg.worksheet 17

18 Evaluation: Practice! pg.897 # f(x) = x+4 x x 4 lim f(x) = x 5 lim f(x) = 2. 18

19 Again! step it up! Left and Right hand limits separately Use when there is a piecewise function or absolute values in the function Because: ex. lim x 3 x 3 ex. lim ( + ) x 0 1 2x 1 2x 19

20 Special Infinity Rules When the denominator has greater degree then... lim 1 x x lim 1 x as x gets bigger, 1/x gets smaller 0, since isn't a number, we can get as close to 0 as we want x k = 0 lim 1 x k = 0 x lim e x x =0 20

21 Special Infinity Rules Example ex. f(x) = 4x 3 x 3x ex. f(x) = 4x 23 x 3x x 20 21

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