2.1A: Rates of Change & Limits Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: If you look at your speedometer at some time during this trip, it might read 65 mph. This is your instantaneous speed at that particular instant. average speed = change in position = Δy elapsed time Δt

A rock falls from a high cliff… The position (measured from the cliff top) is given by: Position at 0 sec: average velocity from t=0 to t=2: What is the instantaneous velocity at 2 seconds? Position at 2 sec:

for some very small change in t where h = some very small change in t First, we can move toward a value for this limit expression for smaller and smaller values of h (or Δt) …

We can see that the velocity limit approaches 64 ft/sec as h becomes very small. We say that near 2 seconds ( the change in time approaches zero), velocity has a limit value of 64. (Note that h never actually became zero in the denominator, so we dodged division by zero.) Evaluate this expression with shrinking h values of: 1, 0.1, 0.01, 0.001, ,

The limit as h approaches zero analytically: = = = =

Consider: What happens as x approaches zero? Graphically: WINDOW Y=

Looks like y→1 from both sides as x→0 (even though there’s a gap in the graph AT x=0!)

Numerically: TblSet You can scroll up or down to see more values. TABLE

It appears that the limit of is 1, as x approaches zero

Limit notation: “The limit of f of x as x approaches c is L.” So:

The limit of a function is the function value that is approached as the function approaches an x-coordinate from left and right (not the function value AT that x-coordinate!) = 2

Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See page 58 for details.) For a two-sided limit to exist, the function must approach the same height value from both sides. One-sided limits approach from only the left or the right side.

The limit of a function refers to the function value as the function approaches an x-coordinate from left and right (not the function value AT that x-coordinate!) (not 1!)

Near x=1: limit from the left limit from the right does not exist because the left- and right-hand limits do not match! = 0 = 1

Near x=2: limit from the left limit from the right because the left and right hand limits match

Near x=3: left-hand limit right-hand limit because the left- and right-hand limits match