2.1

Rates of Change and Limits

Grand Teton National Park, Wyoming

Suppose you drive 200 miles, and it takes you 4 hours.

If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

A rock falls from a high cliff.

The position of the rock is given by:

After 2 seconds:

average speed:

What is the instantaneous speed at 2 seconds?

for some very small change in t

where h = some very small change in t

We can use the TI-89 to evaluate this expression for smaller and smaller values of h.

We can see that the velocity approaches 64 ft/sec as h becomes very small.

We say that the velocity has a limiting value of 64 as h approaches zero.

(Note that h never actually becomes zero.)

The limit as h approaches zero:

Consider:

What happens as x approaches zero?

Graphically:

Numerically:

You can scroll down to see more values.

You can scroll down to see more values.

Limit notation:

So:

not 1

Properties of Limits:

Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.

(See your book for details.)

For a limit to exist, the function must approach the same value from both sides.

One-sided limits approach from either the left or right side only.

1

2

3

4

1

2

At x=1:

At x=2:

1

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4

1

2

At x=3:

1

2

3

4

1

2

The Sandwich Theorem:

By the sandwich theorem:

π