1. LIMITS An Introduction to Calculus

2. LIMITS What is Calculus? What are Limits? Evaluating Limits
Graphically
Numerically
Analytically
What is Continuity?
Infinite Limits
This
presentation
Bring your graphing calculator to class!

3. WHAT IS CALCULUS? Calculus is the mathematics of change
dynamic, whereas earlier mathematics is static
a limit machine
Precalculus
Mathematics
Limit
Process
Calculus
As an example, an object traveling at a constant velocity can be modeled with precalculus mathematics. To model the velocity of an accelerating object, you need calculus.

4. TWO CLASSIC CALCULUS PROBLEMS that illustrate how limits are used in calculus
The Tangent Line Problem
Finding the slope of a straight line is a precalculus problem. .
Finding the slope of a curve at a point, P, is a calculus problem. Q
The limit process: Compute the slope of a line through P and another point, Q, on the curve. As the distance between the two points decreases, the slope gets more accurate.
The Area Problem
Finding the area of a rectangle is a precalculus problem.
Finding the area under a curve is a calculus problem.
The limit process: Sum up the areas of multiple rectan-gular regions. As the number of rectangles increases
(and their width decreases) the approximation improves.
See other examples on pages of your textbook.

5. Limits: An Informal Definition
The limit is taken as the input, x, approaches a specific value from either side. In this case, x is approaching c.
The limit itself is a value of the function, i.e., its y value. In this case, the limit is L.
SO…
If f(x) becomes arbitrarily close to a single number, L, as x approaches c from either side, then the limit of f(x) as x approaches c is L.

6. Limits: An Informal Definition
If f(x) becomes arbitrarily close to a single number, L, as x approaches c from either side, the limit of f(x) as x approaches c is L.
The existence or nonexistence of f(x) at c has no bearing on the existence of In fact, even if , the limit can still exist.
IMPORTANT POINT: -5 5 4 -4
even though f(1) is undefined
f(c) is undefined -5 5 4 -4
even though
f(c) ≠ L

7. Limits: Formal Definition
The ε-δ definition of a limit: ε (epsilon) and δ (delta) are small positive numbers.
means that x lies in the interval (c – δ, c + δ) i.e., 0 < |x – c| < δ ↑ means x ≠ c
means that f(x) lies in the interval (L – ε, L + ε) i.e., |f(x) – L| < ε.
If f(x) becomes arbitrarily close to a single number, L, as x approaches c from either side, then the limit of f(x) as x approaches c is L.
This formal definition is not used to evaluate limits.
Instead, it is used to prove other theorems.
FORMAL DEFINITION: Let f be a function on an open interval containing c (except possibly at c) and let L be a real number. The statement
means that for each ε > 0, there exists a δ > 0 such that if
0 < |x – c| < δ, then |f(x) – L| < ε.

8. We will evaluate limits
Graphically
Numerically
Analytically

9. Evaluating Limits Graphically1. 2.
What is ?
What is ?
(2, 6)
(2, 6)
a parabola
a parabola w/ a hole in it
Once again, the existence of a limit at a given point is independent of whether the function is defined at that point.

10. Evaluating Limits Graphically (cont)3.
This rational function is undefined at x = -1, because when x is -1, the denominator goes to zero.
However, x = -1 is not a vertical asymptote* -5 5 4 -4
(-1, 2)
If you factor the numerator, you get
Canceling like factors, you get
This is a straight line. f(x) is still undefined at x = -1,
so there is a hole in the graph at the point (-1, 2).
Even though f(-1) is undefined, is still 2. *
Rational functions can have point discontinuities as well as vertical asymptotes. If the rational expression can be simplified, the function is still not defined at this value of x, but the discontinuity becomes a point rather than a vertical line.

11. Graph a function with a given limit
Draw the graph of a function such that
Sketch any function that approaches (3, -1) from both directions. Simplest example is a straight line, e.g. f(x) = -x + 4.
Three cases: -5 5 4 -4
(3, -1)
f(3) = -1 -5 5 4 -4
(3, -1)
f(3) is undefined -5 5 4 -4
(3, 3)

12. Limits that do not exist
Limits do not exist when:
f(x) approaches a different number from the right side of c than it approaches from the left side.
f(x) increases or decreases w/o bound as x approaches c.
f(x) oscillates between two fixed values as x approaches c.
We will consider these cases graphically.

13. Evaluating Limits Graphically Limits that do not exist
f(x) approaches a different number from the right side of c than it approaches from the left side. 5.
This function is undefined at x = 0, because the denominator goes to zero.
When x > 0,
-2.5
2.5 2 -2
Approaching zero from the right,
(0, 1)
This is called the right-hand limit.
(0, -1)
When x < 0,
Approaching from the left,
This is called the left-hand limit.
When the behavior differs from right and left, the limit does not exist.

14. Evaluating Limits Graphically Limits that do not exist
f(x) increases or decreases without bound as x approaches c. 6.
This function is undefined at x = 3, because the denominator goes to zero. It can not be simplified, so there is a vertical asymptote at x = 3.
Approaching 3 from the right, f(x) increases without bound.
Approaching 3 from the left, f(x) decreases without bound.
When the function increases or decreases without bound, the limit does not exist.

15. We will evaluate limits
Graphically
Numerically
Analytically

16. Evaluating Limits Numerically
Ex 1: Find
We will do this using the Ti89 and its table function.
Set Graph the function. What happens to f(x) as x
approaches 1 from the left and from the right?
Use the TBLSET function.
Set TblStart = 0.8 and set Tbl = Go to the table.
What happens as you move down the table from x = 0.8 to x = 1?
Now set TblStart = 1.2.
What happens as you move up the table from x = 1.2?
In both cases, the f(x) values are getting closer and closer to 2, therefore

17. Evaluating Limits Numerically
Ex 2: Find
Set Graph the function. What happens to f(x) as x
approaches 3 from the left and from the right?
Use the TBLSET function.
Set TblStart = 2.8 and set Tbl = Go to the table.
What happens as you move down the table from x = 2.8 to x = 3?
Now set TblStart = 3.2.
What happens as you move up the table from x = 3.2?
In both cases, f(x) values are growing without bound as x approaches 3 from the left and from the right. Therefore we say
does not exist

18. LIMITS Limits are a fundamental part of Calculus
The limit of a function at a specific input value, c, is the value of the function as you get increasingly closer to c.
The function need not be defined at this input value, c.
If the function is defined at c, the limit value and function value need not be the same.
Limits can be evaluated
Graphically,
Numerically, and
Analytically
Limits do not exist if,
The function approaches different values from the right and left
The function increases or decreases without bound
The function oscillates between 2 fixed values
Today, we focused on evaluating limits by studying the graphs of functions and numerically approaching x from both directions.
Next we will be evaluating limits analytically.

19. Homework
Pg. 47 (1-8)
Pg. 54 – 55 (6,8, 9-18)