Area of a Plane Region
f(x)
is continuous and nonnegative on interval [a,b]
Area is bounded by f(x), xaxis, x=a, and x=b
However, not all limits exist.
Example:
A limit of a function f(
x
) is the value the function approaches as
x
becomes infinitesimally close to a specified value,
c
.
Direct Substitution
Sometimes, it's as simple as plugging and chugging.
Rationalizing
1. Multiply by the conjugate
2. Simplify
3. Direct substitution
Dividing Out
1. Factor out common factors in the numerator and denominator.
2. Cancel them out.
3. Direct substitution!
Tangent Lines
Difference Quotient
We learned this at the beginning of the year – it's finally time to apply it.
What is a limit?
Solving limits
nonInfinity
Limit Applications
LIMITS
by Emilio ArroyoFang, Bethany Hung, & Brittany Lau: Group 3, 5th period
Q1:
What is the limit L as x approaches c?
2
There are three conditions for which a limit does not exist.
Left & Right Behavior
A function does not have a limit if as x > c, the left side approaches a different value from the right side.
Unbounded Behavior
A function does not have a limit if as x> c, the y value increases or decreases without bound, or without reaching a set value.
Oscillating Behavior
As x > c, f(x) can also oscillate between two values.
Example:
f(
x
) =

x

x
—
Example:
f(
x
) =
1
x
—
2
Note: Oscillation ≠ wave
Example:
f(
x
) =
1
x
—
sin
(
)
What about sine or cosine waves?
There are definitely limits for sin(x) and cos(x), as long as there is a clear yvalue as x>c.
There are three ways to solve for a limit.
Direct substitution
Dividing out
Rationalizing
Use the latter two if direct substitution fails.
lim
x
→5
9
x
+2
= 47
9(5)+2
However, this doesn't always work.
Sometimes, plugging it in will result in something called the
indeterminate form
.
lim
x
→5
x5
=
25
x
2
0
—
0
?????
In that case, you need to use either the
Dividing out
or
rationalizing
techniques.
lim
x
→5
x
5
=
25
x
2
x
5
(
x
5)(
x
+5)
=
1
x
+5
=
1
10
5
example
Note: This technique of solving for the limit is called the
numerical approach
.
example
Properties of Limits
Limits of individual functions can be added together, much like plain numbers.
Limits of Rational Functions
These points tell us how we can try to solve limits.
OneSided Limits
Here, you find the limit as x » c from the
left
and as x » c from the
right
.
Limit from the left
Limit from the right
To constitute as a limit, the values must be the
same
. If not, there is no limit.
The
difference quotient
is widely used in beginner calculus. Here, we'll use it to find the
derivative
of a function at a certain point.
difference quotient
derivative
Q1.
SOLVE.
i.e. derivatives
A
derivative
is the
slope of a function at a certain point
.
The red lines are called
tangent lines
, and the slopes of the tangent lines are the derivatives at the tangential points.
Tangent lines can intersect noncircular graphs at more than one point.
The tangential slope can be estimated from these diagrams by approximating
rise over run
, but this will yield inaccurate results.
Slopes & Limits
Instead of crude approximations, we can use limits to find the tangential slope. To understand this process, we'll make use of
secant lines
.
Here, the
pink
line is a secant line.
What would we call its slope?
The Difference quotient
As
h
approaches 0, the secant line becomes a tangent line.
Thus, provides the tangent line slope.
This resultant slope is called a derivative.
With the linear derivative generated from a quadratic function, you can then calculate the tangential slope of any point on the curve.
Simply substitute the x value of (x, y) into the derivative f(x).
Q2
Find the derivative of
and the tangent line at (5, 110)
Answer 2
Limits at infinity
Now we know how to use limits to solve for slopes and lines. That's onedimensional, but we can get a onedimension upgrade if we use
limits at infinity
.
∞
Limits at infinity are used when solving for .
area
∞
Solving Limits at
If
r
is a positive real number, then
Solving Limits at Infinity for Rational Functions
Limits of Summations
f(x) is rational function f(x) = N(x)/D(x)
To solve for the area under the curve, we'll use limits of
summations
.
As x approaches +/ infinity ...
Example of finding area of plane region
Limit of a Sequence
{
If { } is a sequence and f(n)= , n is a positive integer.
If n > m, the limit does not exist because it is unbounded.
Example
0
As n increases, the sequence converges to 0.
Height
Width
Example of applying Summation formulas to simplify
The Area Problem
Therefore this function has no limit
To do this, we'll need to review basic summation formulas.
=
=
=
=
=
area under a curve
Find the area under the curve of the function
From x=1 to x=3
Q3
You have 1 minute.
Q4
Answer 4
Thanks for watching!
What do organic mathematicians put into their fireplaces?
NAtural logs
=
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