Limits with piecewise functions, One-sided limits, and More Algebra with Limits

We started class today by going over the homework from the night before (4/30). We then started talking about the limits of piecewise functions and one-sided limits, which is when you determine the limit of something by looking at it from either the positive or negative direction. This is indicated by having a + or - superscript. We then had more examples of algebra with limits.

More Limit Examples

 Today in class we went over more examples of limits. We focused mainly on how to approach the problem if  you get the fraction (0/0) when you first "plug in" the limit number.  If you do get (0/0) then you have to factor in order to cancel out like terms.  The "secret" that we learned today, was that you solve derivatives the same way you solve these limit problems. 

When a limit does not exist; Algebra and Limits; Direct Substitution

Today we learned that 3 reasons that a limit does not exist: the limit does not exist because of oscillating behavior; the limit does not exist because of unbounded behavior; the limit does not exist because of different left/right behavior.  We learned that algebra and limits is "nice" because it makes sense to us.  Direct substitution involves plugging the limit number into the equation.

Evaluating Limits

So today we started limits. We use limits as "mathematical microscopes" to describe y-values near a given x-value. We evaluate the limits of given functions by (for now) using the calculator: Enter the function in Y1 and use the table feature to enter certain x-values. Use the y-values to determine the limit of the function. Remember to use radians when evaluating limits that involve trig. Also, keep in mind that some limits do not exist or may be deemed "indeterminate." See notes for further details.