LIMITLESS

1. Limits are the behavior of a function as it approaches an x-value from either side. 

2. When evaluating a limit, first try to plug in the number to see if it works. If the equation equals zero on the bottom, move to the next step.

If it is a square root function, rationalize the limit by multiplying the square root and opposite reciprocal of the constant with the original equation on both the numerator and denominator. Simplify. Then set the x to see if the limit exists. 

If it is a exponential function, factor it to cancel like terms. Simplify.

If it is two fractions, combine them by making the denominators the same. Simplify.

      4. In piecewise graphs the limit may not always exist. If the behavior of the graph from the left and the right do not come to the same point, and one does not exist, then the limit does not exist.

In this graph there is no limit at 1 because the behavior of both sides of the graph do not match up.

      5. Infinite limit: if the denominator is an extremely small integer, the graph will have an infinite limit, or asymptote. If one side of the graph goes to positive infinity while the other goes to negative infinity, the limit does not exist. If they both go in the same direction, the limit is infinite(either positive or negative). 

The graph will not reach 0 by a very small margin, the equation increasing until positive infinity.

Polynomial

y= 7x^3 - 11x^2 - 62x - 24

• The y-intercept can be found by looking at the constant term in the equation, or the number with no x by it. In this equation it's -24.
• To find out how many possible positive, negative, or imaginary solutions there are, use Descartes' Rule of Signs. Write the signs of the original equation and determine how many changes there are.  In this equation, (+ - - -) there is only one change. This means that there is one possible positive solution. The next step is to multiply the equation by -x, which means to change all of the odd exponents' coefficients will be opposite. So (- - + -), the new equation has 2 changes which means there are 2 possible negative or imaginary solutions. (Since imaginary numbers come in pairs.)
• The Rational Root Test provides all of the possible solutions to the problem. To find the rational roots take the constant term and put it over the leading coefficient (the coefficient before the highest degree). (-24/7) Take all of the possible factors of each number and keep them in the fraction. 1,2,3,4,6,8,12,24,-1,-2,-3,-4,-6,-8,-12,-24 / 1,7,-1,-7  This means that any of these solutions can be plugged into synthetic division to be proposed as a solution.
• Synthetic Division is the next step and to do it you must write the coefficient of each of the parts of the equation and the constant, including if there is a 0x. Place a factor you are testing on the outside of these numbers. The coefficients get dropped down and are multiplied by the number outside of the equation, starting from the first to the last. If the last number equals 0, the factor is a correct solution. Take the new numbers that developed from multiplication and apply them to a new rational root. If it does not equal 0, try again with another rational root.
  
• To check the solutions with your calculator, type the equation into the y= and where the line crosses the x-axis, those are your solutions. In this problem they are -2, 3/7, and 4.
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The 3 common trig triangles, their side lengths, and their sine, cosine, and tangent values.