The document defines limits and discusses different methods for solving limits as the variable approaches a number or infinity. It introduces limit notation and explains how to solve limits by substitution, using sum and product laws, and handling undefined and zero cases. When taking a limit as the variable approaches infinity, it provides approaches such as determining the limit based on the highest polynomial term, and whether the numerator or denominator increases faster for rational functions.

1. Limits By Kathleen Hook

2. Limit Definition Notation: limf(x)=L X -> a The Range (Y) Values the function is approaching as the domain (X) values get closer to a

3. Solving Limits When x -> a number Plug the number into the function Lim x -> Lim (a) X-> a X-> a

4. Solving limits Ex) lim (3x) = X-> 4 Substitute the number: lim 3(4) = 12 X-> 4

5. Sum Law Limit of sum = sum of limits ex) lim(x²+2x-5) = X->2 lim x² + lim 2x – lim5 = X->2 X->2 X->2 (2)²+ 2(2) -5 = 3

6. Product Law Limit of product = product of limits Ex) Lim(X+6)(X-4) = X-> 2 lim(x+6) lim (x-4) = X-> 2 X-> 2 (2+6) (2-4) = (8) (-2) = -16

7. Quotient Law Limit of quotient = quotient of limits Ex) lim(x+1) = (x-2) X-> 4 Lim (x+1) = (4+1) = 5 (4-2) 2 X-> 4 _________________ Lim (x-2) X-> 4

8. Solving Limits when Substituting makes the function undefined Undefined: When the bottom is zero Solve to eliminate like terms

9. Solving Undefined ex) lim (x²-16) (x—4) X-> 4 Plugging in 4 will equal zero Factor the top: x² -16 -> (x+4)(x-4) -> (x+4)(x-4) (x-4) Cancel like terms Lim (x+4) = (4+4) = 8 X->4

10. Numerator equals zero The limit is still defined Ex) lim(x-6) = (8-x) X-> 6 (6-6) = 0 = 0 (8-x) 2

11. Solving Limits When X approaches infinity Notation: lim f(x) 8 x -> There are multiple approaches

12. Infinity by Graphing Ex) Lim1 x 8 X-> Graph: Y values approaching zero

13. Limit by Plotting Ex) Lim 1 x *already found the limit is Zero X-> 8 Proof: X f(X) 1 1 5 1/5 F(x) values approaching zero 100 1/100 1000 1/1000

14. Infinity New Way Polynomial Functions: Determined by highest power Even: Ex) limx² = …. = +Infinity Odd: Ex) lim5-3x= … = -Infinity

15. Rational Functions Denominator increase faster lim=0 Ex) lim1 = 0 5x Ex) lim3x= 0 x²+1

16. Rational Functions Numerator increases faster lim = - or 8 8 8 Ex) 4x-8x² = - 2x-1 EX) x²+2x = x+9 8 If Powers Even: ex) lim4x+1= 4 3x-8 3