Unit U Big Question #3

3. How do we evaluate limits numerically, graphically, and algebraically?

NUMERICALLY

When we evaluate limits numerically, it is on a table. We are finding the intended height of the function, which is the y-value as the graph, and it is referred to as the LIMIT. We use a table similar to the one in the picture above. As x is approaching a certain value, in this case 3, we write numbers approaching the limit from the left and from the right. This way, we ALWAYS find what the limit is intending to reach even if the graph never actually reaches that point. (Image from Unit U SSS Packet)

GRAPHICALLY

(Images from www.education.com)

When we evaluate limits graphically, we are looking at a graph and using our fingers to approach the limit from the left and right to see if they meet.

ALGEBRAICALLY

There are three ways of evaluating limits algebraically: direct substitution method, dividing out/factoring method, and rationalizing/conjugate method.   

1.  Direct substitution is the method used first. We take the limit and plug in that number into all the x's in the equation to find an answer. We can get a numerical answer which is what we hope for and we are finished. We can get 0/# and this means our limit is 0 and we are finished. We can get #/0 which means the limit is undefined and DNE because of unbounded behavior. And lastly, we can get 0/0 which means we are not finished. 0/0 is indeterminate and we need to use one of the other methods to evaluate it.

2. Dividing out/factoring method is used after we get indeterminate form (0/0).  We factor the numerator and denominator and cancel any common terms on top and bottom. When we do this, we are removing the zero in the denominator. Then, we use direct substitution with the simplified expression. This is similar to finding a hole in a past unit.

3. Rationalizing/conjugate method is used when the fraction has a radical in it. We use the conjugate of either the numerator or the denominator depending on where the radical is. We foil the conjugates and leave the non-conjugate factored. Then, we use direct substitution.