1 2 finding limits graphically and numerically
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1.2 Finding limits graphically and numerically. Calculus has its limits. Objective:. To solve limits numerically and graphically To analyze properties of limits. What is calc?. Calc is the math of change– of velocities and accelerations Pre-Calc Calc Static Dynamic

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Presentation Transcript

Objective
Objective:

  • To solve limits numerically and graphically

  • To analyze properties of limits


What is calc
What is calc?

  • Calc is the math of change– of velocities and accelerations

  • Pre-Calc Calc

  • Static Dynamic

  • Constant velocity Velocity of accelerating objects

  • Slope of a line Slope of a curve

  • Area of a rectangle Area under a curve


What is calc1
What is calc?

  • Calc is a limit machine

  • Pre-calc  Limits  Calculus

  • 2 main problems:

  • Tangent line problem (Ch 2)

  • Area problem (Ch 4)


Intuitive approach to limits
Intuitive approach to limits

  • A limit is an expected value

  • The actual value may be different or undefined

  • Actual value at x=1: Expected value at x =1:


More formally
More formally…

  • If f(x) becomes arbitrarily close to a single number L as x approaches C from both sides, the limit of f(x) as x approaches c is L.

  • Written as:


In other words
In other words

  • Three things must happen for a limit to exist

  • The limit from the right exists

  • The limit from the left exists

  • The limit from the right equals the limit from the left


Ways to look at calc
Ways to look at calc

  • Graphically

  • Numerically

  • Algebraically


Graphically
Graphically

  • To find limits, use a graph.


Numerically
Numerically…

  • Make a chart

    at x= 2


Another example
Another example

  • at x = 1


Graph by hand
Graph by hand

  • What is the limit at x=2



Why limits do not exist dne
Why limits Do Not Exist (DNE)

  • 3 main reasons

  • 1. f(x) approaches different numbers from the right and left

  • 2. f(x) increases or decreases without bound

  • 3. f(x) oscillates between fixed values


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