Scribe Post 9-14

Hi everyone! We started class off today by opening the floor for questions on old IWs or concepts that we didn’t understand, which according to OB is a rare opportunity.

Pg. 66 # 31 

Solve algebraically:

Pg. 76 #7

Find the horizontal asymptotes of  f(x)= x/|x|
*You can look at this graphically or numerically, but essentially you're looking for the following:




Numerically, you let x equal a very large number or very small number and plug it into the equation.
Graphically, you graph the function and can tell visually what the asymptotes are. The graph would be the following:

As you can see, the asymptotes are y=1, as x approaches infinity and y=-1 as x approaches negative infinity.

Holes vs. Asmyptotes 

The previous question led to a question of "how do you differentiate between a hole and an asymptote?" This led to a half hour discussion that OB claimed was the "best use of half an hour". We came to the conclusion that:

*If non zero on top and zero on bottom= asymptote
*If zero on top and zero on bottom= hole 
*** Make sure your functions are factored first, or this may not hold true! 

Of course, we then completed a few examples.

This would have an asymptote, because the numerator remains a non zero number, but the denominator will be zero when x equals zero.



 This would have a hole, because when x equals zero, both the numerator and denominator will be zero.

Continuity: 

Definitions-

• Continuous point: On a function, f, a point c is continuous if

              In other words, if the limit and y value are the same, it’s a continuous point

• A function is continuous if it is continuous at all points on its domain  
• Graphisthenics; rational functions (polynomial/polynomial) are continuous because the holes and asymptotes are not on the domain 
• All six trig functions are continuous, so are their sums, differences, products, quotients and compositions
• For endpoints, continuity is determined by a one-sided limit
• Staircase function is NOT continuous

We looked at  the following function to determine which points are continuous;


Non-continuous points [-1,4]
1 (removable)
2 (Jump)
3 (Infinite)
-1(removable)






We then began to question, what is removable discontinuity? Jumping? Infinite? OB then gave us some more definitions.

Definitions (set two)

• Removable discontinuity: If you can redefine the point and make it continuous, then it’s removable 
• Exists when the limit of the function exists, but one or both of the other two conditions is not met
• Jump discontinuity: non-removable, see x=2 on the above example for a visual 
• Exists when the two-sided limit does not exist, but the two one-sided limits are both finite, yet not equal to each other
• Infinite discontinuity: non-removable asymptote, see x=3 
• This exists when one of the one-sided limits of the function is infinitelimx→c+f(x)=∞
• Oscillating discontinuity: f(x)=sin(1/x) - barcode function 
• exists when the values of the function appear to be approaching two or more values simultaneously

We were all told to look up what the barcode function looks like, but it is also listed here










An example of a non-continuous function would be:



We were asked to speculate what this function would like like, and as Francie pointed out, it looks "redic" Mathematicians apparently refer to these sort of functions as "pathological" because of their counterintuitive behavior. 




This is what it would look like (image from milefoot.com). Because the values bounce between rational and irrational, they are not continuous, because limits do not exist for each point.





In class we had different conditions, of y=1 when x is rational, and y=-1 when x is irrational. However, the above problem may be on a quiz or test as a bonus! *Hint hint*








For clarification on domain, holes and asymptotes, watch this link here.
For clarification on continuity, watch part one and part two of the Patrick JMT videos.



The scribe for next class is Autumn! (Good luck!)