in this section, we will investigate the idea of the limit of a function and what it means to say a...
TRANSCRIPT
In this section, we will investigate the idea of the limit of a function and what it means to say a function is or is not continuous.
Section 2.3 Limits and Continuity
DefinitionInformal
The limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to a.
That is, the closer x gets to a (from either side), the closer y will get to L.
Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.
Graphically
Consider the graph below of y = f(x).
Graphically
Consider the graph below of y = f(x).
Numerical Ex. 1
Consider the function .
Numerical Ex. 1
Consider the function .
x f(x)
1.9 4.9
1.99 4.99
1.999 4.999
x f(x)
2.1 5.1
2.01 5.01
2.001 5.001
Numerical Ex. 1
Consider the function .
x f(x)
1.9 4.9
1.99 4.99
1.999 4.999
x f(x)
2.1 5.1
2.01 5.01
2.001 5.001
Numerical Ex. 2
Consider the function .
Numerical Ex. 2
Consider the function .
x f(x)
3.9 -0.9
3.99 -0.99
3.999 -0.999
x f(x)
4.1 1.1
4.01 1.01
4.001 1.001
Numerical Ex. 2
Consider the function .
x f(x)
3.9 -0.9
3.99 -0.99
3.999 -0.999
x f(x)
4.1 1.1
4.01 1.01
4.001 1.001
DefinitionInformal
The right hand limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to, but larger than, x = a.
That is, the closer x gets to a (from from the right), the closer y will get to L.
Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.
DefinitionInformal
The left hand limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to, but smaller than, x = a.
That is, the closer x gets to a (from from the left), the closer y will get to L.
Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.
Graphically
Consider the graph below of y = f(x).
Graphically
Consider the graph below of y = f(x).
Numerically
Consider the function .
x f(x)
3.9 -0.9
3.99 -0.99
3.999 -0.999
x f(x)
4.1 1.1
4.01 1.01
4.001 1.001
Numerically
Consider the function .
x f(x)
3.9 -0.9
3.99 -0.99
3.999 -0.999
x f(x)
4.1 1.1
4.01 1.01
4.001 1.001
Theorem
if and only if
and
So both left and right hand limits must agree for the overall limit to have a value.
Graphical Example
Use the graph of y = f(x) below to find:
Algebraic Example 1
Algebraically find
Algebraic Example 2
Algebraically find
Derivative Example 1
Use the definition of derivative to find for the
function
Definition
A function f is continuous at x = a if .
So both the left and right hand limits must exist, the function itself must exist, and all three of these must be equal to each other.
Graphical Example
Where is f not continuous? Why?
Algebraic Example 3
Consider the function
Is f continuous at x = 4? Support your answer.
Algebraic Example 4
Find all values of “a” so that
is continuous for all values of x.