1. What is a limit?

2. A Geometric Example 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.

3. Numerical Example 1
Let’s look at a sequence whose nth term is given by:
What will the sequence look like?
½ , 2/3, ¾, 5/6, ….99/100, 99999/100000…

4. What is happening to the terms of the sequence?
½ , 2/3, ¾, 5/6, ….99/100, 99999/100000…
Will they ever get to 1?

5. Graphical Examples

6. Graphical Example 1
As x gets really, really big, what is happening to the f(x)?

7. The Derivative as the Slope of the Tangent Line

8. What is derivative?

9. Cow Milk Derivatives

10. What is a derivative?
A function
the rate of change of a function
the slope of the line tangent to the curve

11. The Tangent Line
(we’ll try to find the slope/gradient of the tangent line using the idea of limit)
Note: tangere = to touch.
y-axis
single point
of intersection
x-axis

12. Slope of a Secant Line
Let’s begin with finding the slope of a secant line (secare: to cut)
y-axis
f(a) - f(x)
a - x
f(x)
f(a) x a
x-axis

13. Slope of a (closer) secant line
y-axis
f(a) - f(x)
a - x
f(x)
f(a) x x a
x-axis

14. Closer and closer…
y-axis a
x-axis

15. Watch the slope...
y-axis
x-axis

16. Watch what x does...
y-axis x
x-axis a

17. The slope of the secant line gets closer and closer to the slope of the tangent line...
y-axis
x-axis

18. What happen as the values of x get closer and closer to a???
y-axis x a
x-axis

19. The slope of the secant lines gets closer
to the slope of the tangent line...
...as the values of x
get closer to “a”
Translates to….

20. f(a) - f(x)
lim
a - x x a
Equation for the slope
as x approaches a
Which gives us the the exact slope of the tangent line to the curve at point a….

21. Similarly... f(x) f(a) x a h x+h y-axis f(x+h) – f(x) (x+h) – x
x-axis h
x+h

22. thus...
lim f(x+h) – f(x)
h 0 h
Give us a way to calculate the slope of the tangent line at point a.

23. A simple example...
want the slope
Where x=2

24. and the gradient at x = 2 is 4

25. furthermore... The gradient function is 2x This is the one called
“the derivative”
and the symbol is
f’(x) or dy/dx
When x=2,
the slope is 4

26. What is a derivative?
A function
the rate of change of a function
the slope of the line tangent to the curve

27. in conclusion...
The derivative is the the slope of the line tangent to the curve (evaluated at a point)
it is a limit
once you learn the rules of derivatives, you WILL forget these limit definitions (some people call it first principle)