1. Finding Limits Graphically and Numerically
Lesson 1.2
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2. What is a limit?
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3. A Geometric Example Look at a polygon inscribed in a circle
As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.
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4. If we refer to the polygon as an n-gon,
where n is the number of sides we can make some mathematical statements:
As n gets larger, the n-gon gets closer to being a circle
As n approaches infinity, the n-gon approaches the circle
The limit of the n-gon, as n goes to infinity is the circle
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5. The symbolic statement is:
The n-gon never really gets to be the circle, but it gets close - really, really close, and for all practical purposes, it may as well be the circle. That is what limits is all about!
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6. Numerical Example Let’s look at a sequence whose nth term is given by:
What will the sequence look like?
½ , 2/3, ¾, 5/6, ….99/100, 99999/100000…
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7. What is happening to the terms of the sequence?
½ , 2/3, ¾, 5/6, ….99/100, 99999/100000…
Will they ever get to 1?
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8. Let’s look at the sequence whose nth term is given by
1, ½, 1/3, ¼, …..1/10000, 1/ ……
As n is getting bigger, what are these terms approaching?
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9. Common Types of Behavior Associated with Nonexistence of a Limit
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10. Definition of Limit
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11. 4/8/2019

12. Finding Limits
EXAMPLE
Determine whether the limit exists. If it does, compute it.
SOLUTION
Let us make a table of values of x approaching 4 and the corresponding values of x3 – 7 as we approach for both from above (from the right) and below (from the left)
x x3 - 7
As x approaches 4, it appears that x3 – 7 approaches 57. In terms of our notation,
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13. One more try….. Turn on the TI-83/84 or 89 Graph y = 2x + 2
Create a table
Study what happens as x approaches 5.
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14. From the Left
From the Right x
f(x) x
f(x)
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15. Example. Evaluate the following limit:
Let’s make a table of values…
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16. From the Left
From the Right x
f(x) x
f(x)
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17. What causes this discontinuity??? ALGEBRA!
Graphically!
What causes this discontinuity???
ALGEBRA!
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18. Simplify:
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19. Asymptotes Vs. Holes Set the denominator equal to zero.
The solutions to this will show you where there is some type of discontinuity.
When a factor cancels out in the numerator and denominator, this cause a hole discontinuity.
When there is no cancellation, the resulting solution leads to a vertical asymptote.
What about horizontal asymptotes??? Hold your pants, they are coming soon.
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20. When we are computing limits the question that we are really asking is what y value is our graph approaching as we move on towards x = 2 on our graph.
We are NOT asking what y value the graph takes at the point in question!
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21. Example. Evaluate the following limit:
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22. The limit is NOT 5!!! Remember from the discussion after the first example that limits do not care what the function is actually doing at the point in question. Limits are only concerned with what is going on around the point.
Since the only thing about the function that we actually changed was its behavior at x = 2 this will not change the limit.
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23. Finding Limits
EXAMPLE
For the following function g (x), determine whether or not exists. If so, give the limit.
SOLUTION
We can see that as x gets closer and closer to 3, the values of g(x) get closer and closer to 2. This is true for values of x to both the right and the left of 3.
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24. Limit of the Function Note: we can approach a limit from
left … right …both sides
Function may or may not exist at that point
At a
right hand limit, no left
function not defined
At b
left handed limit, no right
function defined a b
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25. Observing a Limit
Can be observed on a graph.
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26. Observing a Limit
Can be observed on a graph.
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27. Observing a Limit Can be observed in a table
The limit is observed to be 64
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28. Find each limit, if it exists.
In order for a limit to exist, the two sides of a graph must match at the given x-value.
D.S.
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29. The “Wall” Method:
As an alternative approach to Example 1, we can draw a “wall” at x = 1, as shown in blue on the following graphs. We then follow the curve from left to right with pencil until we hit the wall and mark the location with an × , assuming it can be determined. Then we follow the curve from right to left until we hit the wall and mark that location with an ×. If the locations are the same, we have a limit. Otherwise, the limit does not exist.
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30. 1.1 Limits: A Numerical and Graphical Approach
Thus for Example 1:
does not exist
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31. Non Existent Limits
Limits may not exist at a specific point for a function
Set
Consider the function as it approaches x = 0
Try the tables with start at –0.03, dx = 0.01
What results do you note?
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32. Non Existent Limits
Note that f(x) does NOT get closer to a particular value
it grows without bound
There is NO LIMIT
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33. Non Existent Limits
f(x) grows without bound
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34. Non Existent Limits
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35. What happens as x approaches zero?
Graphical Example 2
What happens as x approaches zero?
The limit as x approaches zero does not exist.
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36. What happens as x approaches zero?
Graphical Example 2
What happens as x approaches zero?
The limit as x approaches zero does not exist.
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37. Example. Evaluate the following limit:
Let’s use a table again.
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38. Now, if we were to guess the limit from this table we would guess that the limit is 1. However, if we did make this guess we would be wrong. Consider any of the following function evaluation…
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39. In all three of these function evaluations we evaluated the function at a number that is less that and got three totally different numbers. 
Recall that the definition of the limit that we’re working with requires that the function be approaching a single value (our guess) as t gets closer and closer to the point in question.  It doesn’t say that only some of the function values must be getting closer to the guess.  It says that all the function values must be getting closer and closer to our guess. 
To see what’s happening here a graph of the function would be convenient. 
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40. From this graph we can see that as we move in towards t=0 the function starts oscillating wildly and in fact the oscillations increases in speed the closer to t=0 that we get.  Recall from our definition of the limit that in order for a limit to exist the function must be settling down in towards a single value as we get closer to the point in question.
This function clearly does not settle in towards a single number and so this limit does not exist!
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41. Limits that fail to exist
When the limit approaches different values from the right than from the left, later we call this jump discontinuity
When the graph goes to infinity or negative infinity, ie: a vertical asymptote, later we call this infinite discontinuity
Oscillating behavior – the limit does not exist
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42. SUMMATION Introduction to limits
The limit of a function is the y value the graph is getting closer to as x gets closer to a particular value
Making a table of values to calculate the limit – must be done on a calculator
Sketch a graph to calculate the limit, or use an already existing graph to calculate the limit
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43. Graphically, what is a limit?
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