1.2 Finding limits graphically and numerically

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 • Constant velocity Velocity of accelerating objects • Slope of a line Slope of a curve • Area of a rectangle Area under a curve

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 • 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… • 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 • 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 • Graphically • Numerically • Algebraically

Graphically • To find limits, use a graph.

Numerically… • Make a chart at x= 2

Another example • at x = 1

Graph by hand • What is the limit at x=2

Are there always limits?

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

1.2 Finding limits graphically and numerically