Blog Post 3: Limits

1. A limit is the behavior of a function as it approaches an x-value from either side.

2. You can evaluate a limit by plugging an x-value into a continuous function. To determine whether or not a function is continuous, you can plug in a number greater than and less than the x-value to see if the two sides of the function match up.
     If you plug the x value into the equation and the denominator is zero, then you need to manipulate the function and try again. You can:
 rationalize:

You can get rid of a radical by multiplying the fraction by the conjugate and simplifying to cancel then plugging in the x-value to determine the limit.

 factor:

 

You can factor the numerator and denominator in order to cancel one of the factors and then plug in the x value to find the limit.

 combine fractions:

You need a common denominator to combine the fractions, then you can use that fraction to determine the limit. 

3. If a function is not continuous (such as with a piecewise) then limits will not always exist.

In this piecewise function, the limit of -2 does not exist, even though the function exists at -2. The two sides of the function do not match up at -2, therefore the limit of -2 does not exist.

4. When the denominator is an extremely small positive or negative number, the limit will be infinite. If one side of the function goes to positive infinity while the other goes to negative infinity then the limit does not exist. If both sides go to either positive or negative infinity, then the limit is infinite.

Since both sides of 0 cause the function to be divided by a very small positive number, so the limit of the function at 0 is positive infinity.

Blog Post 2

f(x)= 6x⁴+5x³-65x² +50x+24

The y-intercept is the place where the graph crosses the y-axis, and its ordered pair is (0,24) because the constant term is positive 24.

Descartes is a process used to determine the number of possible solutions (place where the graph crosses the x-axis) of a polynomial. Since the degree (largest exponent) of the polynomial is 4, there are four solutions.

To implement Descartes, you count the number of times the signs in the equation change to determine the number of possible positive rational roots.
     In this example, the sign changes twice, so there are two possible positive rational roots.

In order to determine the number of possible negative rational roots, plug in negative x and then count the number of times the signs change.
     In this example, the equation for negative roots will be: f(x)=6x⁴-5x³-65x²-50x+24. The signs change twice, so there are two possible rational negative roots.

Imaginary roots come in pairs, so it is possible to have no rational roots, no positive rational roots, no negative rational roots, or all rational roots.

The next step to finding the roots is to implement the rational root test, which involves dividing all of the factors of the constant term by the factors of the leading coefficient:

    ±1,2,3,4,6,8,12,24
    _______________      = ±1,2,3,4,6,8,12,24,3/2,1/3,2/3,4/3,8/3,1/6

           ±1,2,3,6

This list represents all of the possible rational roots of this polynomial.

Synthetic division can be implemented to test these roots. Synthetic division involves writing all of the coefficients (including ones that do not appear in the polynomial, such as if there is a 0x not shown), then placing the factor you are testing on the ledge. The leading coefficient gets carried down then multiplied by the ledge number. The product is added to the next coefficient, then that sum is multiplied by the ledge and so on. If the last number ends up as a zero, then the ledge number is a factor.

       2|      6     5     -65     50     24
                      12     34     -62   -24
      -4|      6    17     -31    -12     0
                      -24     28     12
     3/2|     6    -7       -3       0
                       9        3
    -1/3|     6     2        0 
                      -2
                 6    0
The solutions can then be checked by plugging the polynomial into the y= page of a graphing calculator and looking for the ordered pairs in the table. The ordered pairs for these solutions are: (2,0), (-4,0), (3/2,0), and (-1/3,0).



Triangle 1: sin 45°= (sq rt 2)/ 2
cos 45°= (sq rt 2)/2
tan 45°=1

Triangle 2: sin 30°=1/2
cos 30°=(sq rt 3)/2
tan 30°= (sq rt 3)/3

Triangle 3: sin 60°=(sq rt 3)/2
cos 60°=(1/2)
tan 60°= sq rt 3