12.2 Techniques for Evaluating Limits

There are many techniques for finding the limit of a function. Some of these techniques involve algebra, while others involve using a calculator.  Some of the different techniques we will look at are as follows:

1. Direct Substitution
2. Dividing Out
3. Rationalizing

We will also look into:
1. Using Calculators
2. Evaluating One Sided Limits
3. Evaluating Limits from Calculus

Direct Substitution

The first technique for evaluating limits is by using Direct Substitution. This method involves substituting the number theta into your function. For example:







So in this case, the limit of this function is -4. However, the situation changes if the function you begin with is a fraction. This is because if you use the direct substitution method, you can get zeros for your limits which is not correct. In this case, you would need to use the next method: Dividing Out.

Dividing Out

 This technique is very useful when dealing with polynomials. In this method, you will use the fraction to your advantage by dividing out some of the zeros. Here is an example:

 Because we could factor the polynomial in the numerator, we were able to have a common factor that divided out. This left us with a simple subtraction problem where all that we needed to do is plug in for x and solve. If you are given a less friendly equation to begin with, another method used to find limits is called the rationalizing technique.

Rationalizing

This method may be helpful if you are given a problem with a root. To start the rationalizing process, you must multiply both the numerator and the denominator by the conjugate. After this, simplify the equation, and then use the direct substitution method to evaluate the limit. Example:

 Now that you know the limit is not 0, you can proceed to use the direct substitution method...


 

The limit of this function is 1/6. As you can see, rationalizing is a good way to evaluate a limit when you have a root involved.


Using Calculators

Using a calculator can sometimes be a very fast way to deal with limits. The calculator enables you to make a graph of the function, see the table from that graph, or even make your own table. If the direct substitution method produces an indeterminate limit, the graphing calculator will give the correct limit.


Evaluating One Sided Limits

Here is an example of a one sided limit:

For this scenario you must evaluate the limit from both sides of the graph, coming from the right and the left. If you are evaluating the right side of the graph, the limit notation will have a small + sign, and from the left it will have a small - sign. To evaluate this limit, you use the same methods as before, but you must evaluate both sides of the graph.

Evaluating Limits in Calculus

Unfortunately, the fun with limits continues in Calculus :(  It may be seen while using the difference quotient, like this example:

 

Have fun with limits. Wow it's late. I think I'll go to bed now.

12.1 Intro to Limits

Definition of Limit

If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L.

Official Notation:

Simplified Definition:

As x get closer to a point from both side, what would the y or f(x) value most likely be without considering what it actually is.

The limit is what the approaching numbers imply, NOT the actual value.

Solve by Graphing:

Function  

Limit

By looking at the graph, as x approaches the value 1, the line gets closer to the y value 6.

Answer

Types of limits

A limit can either exist or not exist with a graph. Existing one may look a lot like the one above with a few varieties. Non-existing limits can be separated into two categories, one that don't exist and infinities.

A limit as x approaches 2 wouldn't exist.
Either side of the limit approach different numbers.

A limit at 1 could be two things.
(depending on strictness of definition)
the answer could be infinity or does not exist.

With limits that have not definite infinity, they lye on the line between existence. The definition says that it is a number, and since infinity is not a single number, it doesn't exist. But infinity is also true and is useful in calculus.

Solve with a Table:

This process involves plugging in number extremely close to the limit. If both sides of the limit get extremely close to a number, that number is the limit.

10.7 Graphs of Polar Equations

There are a couple different methods one can use to graph polar equations.
 
Point Plotting
To show how to use this method we will use the example equation of:

Because the function of sine repeats itself a full range of values can be found simply by plugging values on the range:

By plugging in these values you should find you get the following values for each angle:

But simply plotting those points you find that the graph is simply a circle with a radius of 2 and and center at (0,2)

To check the graph above is correct you can either convert the Polar Equation to Rectangular Form or graph it using a graphing calculator (Either in polar or parametric mode)

Symmetry
Symmetry is a strategy used as a shortcut of the point plotting method. By recognizing that a graph is symmetrical with respect to either the polar axis, the pole, or the line 

To test if the equation is symmetrical in any of the three ways, substitute the original polar coordinates for those coordinates that would result from a translation across the respective axis or line.
The graphs of each type of symmetry are shown below:

Symmetry with respect to the Polar Axis

Symmetry with respect to the line of

Symmetry with respect to the Pole

If symmetry is discovered fewer points must be plotted to be able to draw the entire graph. Depending of the type of symmetry you can determine which range of points to start with.

Note: Although our first example graph is symmetrical the substitutions will not show so. Thus, we must remember two quick tests for symmetry
1.The graph of the equation below is symmetric with respect to the line

2. The graph of the equation below is symmetric with respect to the polar axis



Zeros and Maximum r-Values
Knowing both were r=0 and the maximum value of r can be valuable pieces of information when graphing. There are no special ways to find the maximum values of r without using your calculator. However, using a calculator you can use either the trace or table features to determine the maximum value of r. Keep in mind that zeros and maximums may be found at more than one point.

Special Polar Graphs
Limacon:

  

More details are in the link below

Rose Curve:
 

n petals if n is odd
2n petals if n is even 

Circle:

Lemniscate: 
 

Click on the link below for more in-depth look at special polar graphs
MORE

10.6 Polar Coordinates

10.6 Polar Coordinates

The polar coordinate system is different from the rectangular system, which is what our class is used to thus far. The system starts at point "O", which is the origin, but in the polar system it is called the "pole". Points in the polar coordinate system are represented in the form of (r,θ), where r (radius)= distance from the pole to the point (P) and θ= the angle deviated from the polar axis.

 A polar coordinate system graph looks like this.

To plot a given point on the polar coordinate system:

1. Start at "O" or the pole

2. Jump r number of rings along polar axis

3. Move θ around the circle

The same coordinate can be represented in an infinite amount of ways. The radius can be positive or negative. 

 or

Converting coordinates from rectangular to polar is simple. To visualize, the polar axis is relatable to the x axis and the pole is relatable to the origin. 

           

   and

                    

When converting entire equations, replace the x, y, r, or θ with is corresponding conversion.

Solve for θ:

Take the inverse tangent of -1 to get:

Solve for r:

Final polar coordinate:

___________________________

Solve for x:

Solve for y:

Final rectangular coordinate:

The polar coordinate system is very relatable to the Unit Circle. When the radius is 1, it is the unit circle. Positive angles for θ always  go counterclockwise. Negative angles for θ always go clockwise.