2.1 FINITE LIMITS One Sided Limits, Double Sided Limits and Essential Discontinuities Mathgotserved.com

Geometric View of Limits Look at a polygon inscribed in a circle As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.

If we refer to the polygon as an n-gon, where n is the number of sides we can make some mathematical statements: As n gets larger, the n-gon gets closer to being a circle As n approaches infinity, the n-gon approaches the circle The limit of the n-gon, as n goes to infinity is the circle The symbolic statement is: Note: The n-gon approaches a circle in appearance even though it is not a circle. It might as well be a circle.

Definition (verbal) The limit of a function is the value the function approaches as the independent variable approaches a value, negative or positive infinity. Or think of it as the y value the function approaches as you approach a specific x value. The Limit of a Function Analytical Representation “Read as the limit of f(x) as x approaches a is L” Can also be expressed as f(x) →L as x→a

Properties of Limits 1. Sum Rule: 2. Difference Rule: 3.Product Rule: 4. Quotient Rule: 5. Constant Rule: 6. Power Rule

WHY LEARN LIMITS? Limits are used in derivatives when mathematicians are calculating the velocity of an object in flight. Real World Application : Calculating Speed(Mathematicians: k/yr)

One-Sided Limits (Verbal) Numbers x near c fall into two natural categories: those that lie to the left of c and those that lie to the right of c. We write [The left-hand limit of f(x) as x tends to c is L.] to indicate that as x approaches c from the left, f(x) approaches L. [The right-hand limit of f(x) as x tends to c is L.] to indicate that as x approaches c from the right, f(x) approaches L We write

How close can Raul Get to the garage?

10www.mathgotserved.com Definition of a Limit Left hand Limit: We say the limit as x approaches a from the left isL Right hand Limit: Double sided Limits L We say the limit as x approaches a is L

Procedures for Finding Finite Limits Algebraically

12www.mathgotserved.com Finding Finite Limits Algebraically Determine the limit of the following

Problem 4: Determine the limit Solution:

Problem 5: Determine the limit Solution:

Procedure for Finding Graphical Limits Point: Find the y coordinate of the point that you are approaching from that direction. Horizontal Asymptote: Find the y coordinate of the equation of the line Vertical Asymptote: One sided +/- infinity, Double sided does note exist.

Problem 1: From both sides you are approaching the point (1,0). The limit is the y coordinate so Solution

Finding Limits Graphically Problem 2: Use the graph to find the following

f(2) = f(4) = Problem 3 dne und 1 2 2

Finding Finite Limits Numerically Problem 1: Find the following numerically using the table of values xf (x)

Problem 2: Find the following limits Numerically xf (x) Error

Problem 3: Find the limit

Problem 4: Find the limit