Our class objective is to "develop an intuitive understanding of limits, including one sided and non-existent limits."
We branched quickly into a warmup where we looked at different equations for limits, and were supposed to try to scrape an answer together. All of these were bad and wrong so we quickly branched into something different.
Taking the first equation of 
we went into geogebra to look at the function. The graph of this function looks like:
There is clearly a large gap in the function at x=1. This makes sense, because in the expression the denominator of (x-1) is zero and the numerator of the expression also equals zero. Rewriting the expression, we came up with a simpler form:
 |
| geogebra messed it up- x≠1, so for the bottom it's when x>1 |
When graphed, the piecewise expression looks like:
(On this graph, the blue is the original function and the black is the subsequent parts from which the piecewise function is made.)
As x approaches 1, the expression has no limit because there is a gap in the graph of the expression. However this does not mean that there are no limits for the expression; there are limits as x approaches other values that are not 1.
The only difference shows up in the graph. Using sliders (k,g(k)), we can see that a point does exist at the function g at 1, however the function f does not have a point there. This can be shown by making a second slider of (k,f(k)), where at 1 there is no point. However, this point does not prove that there is or isn't a limit. The limit is still undefined as x approaches 1. The only difference between the two functions is that at x=1 f(1) does not exist and at g(x) there is a point at g(1).
We then looked at the equation h(x) and the piecewise function p(x). At x=1, there a hole in h(x), however in p(x) the value is 2. Even though there is a missing point, the limit value is 3. The function value doesn't exist, the graph has a hole, but the limit value is 3. The functions are shown in the graph below.
Nonexistent Limits: The easiest way to show a limit does not exist is a graph. For an example, we can use the equation:
The denominator is 0 when x is replaced by 2, so we can't use the substitute to find the limit. From either side, the absolute values of the function values get very large, suggesting that the limit does not exist. As x approaches either an asymptote or a gap, the limit does not exist.
The limit also does not exist when the left hand and right hand limits both exist and are not equal. If the two limits are different, the limit does not exist.
We went over the rules for limits briefly towards the end of the class, which can be found both on page 61 of the textbook and also at this website: <http://www.analyzemath.com/calculus/limits/properties.html>
For the IW #3: Quiz on Iw 1, 2, and 3 next class.
UPDATE: Later in the month, we learned more about when things don't exist, derivatives in particular. Derivatives don't exist at cusps, corners, vertical tangents, and discontinuities. Now we know where limits AND derivatives don't exist.
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