Blog for November 4

Overview:
This week we began learning about limits. We had seen limits before, at the beginning of the derivatives unit. The limit definition of the derivative of a function at a point helped us find the slope of the tangent line at x=c. The limit definition of the derivative of a whole function was used to find the slope of a secant line between any point on f(x) and another point that is h units away from it.
As we move away from the topic of derivatives, we focused on limits as x approaches any point in a function.

Graphical Approach
There are three main approaches to limits. We began with the graphical approach. This entails using graphs to find the limit at a point.


We were asked to find the limits from the left and right of each point pictured, as well as the actual limit for each value.


We learned that the limit does not exist if the limits approaching from the left and right sides are different from each other. The limit also does not exist if either of these values is infinity or negative infinity. In other words...


We then did some practice finding limits using graphs.



Numerical Approach
The next approach is the numerical approach. For this approach, we use a table to determine the limits approaching a point from the left and right. As the value gets closer to x, the limit gets closer to what it should be. For example:

In this example, the limits on either side seem to be approaching the same number, .333. This means the limit as x is approaching 2 is .333. If the limits on  either side are not approaching the same number, or they are approaching infinity or negative infinity, the limit does not exist.


Analytical Approach
The last approach is the analytical approach. This is when you use direct substitution to find the limit. You plug in the value, c, into the function, f(x). For normal functions, if the limit as x approaches c exists, it should be f(c).
 
In the last example, the limit ends up being 0/0. This means that there is a hole in the graph at this point.



We were also introduced to the properties of limits in class, which are listed below.


The last topic we covered this week was limits at infinity. This involves plugging infinity into functions as the x-value,  which would result in finding the end behavior of functions. To find the end behavior or a horizontal asymptote, you need to find the limit as x approaches both infinity and negative infinity.
 

Reminder: Monday's class will be a review for our quiz on Thursday.