Post on 08-Jul-2020
1.5 Limits FI.notebook
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September 24, 2015
1.5 Limits (Welcome to Calculus)
Given:
Translate: "What is f(6)?"
What is the value of the function f when x = 6?
Translate: "What is ?"
What value is the function approaching as x gets very close to 6?
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Algebra studies what is.
Calculus studies what is happening!!
If I want the value of y for a given x, I evaluate the function, ie f(6). This gives me a point on the curve.
The limit describes the value that y approaches as we approach a certain x value. (And we approach from very, very close with teenytiny steps!!)
A limit exists at x = c when a function's value (y) approaches the same number as x gets closer to c from both the left and right.
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Perhaps a visual will assist you in this concept.
From a previous lesson.
I will graph the original in my calculator:
It looks like an unbroken line, but WE know there is a hole. The table will show this.
So we know that f(1) is UNDEFINED!!
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But what value of y is the line approaching as we get closer to x = 1 from the left and right?
(What is ?)
The table will help again, but now the increments of x will be smaller (much smaller)
What value is y approaching from the left?
From the right?
So I'll ask again, what is ?
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In order for a limit to exist at a particular xvalue, the graph must converge (approach) to the same y value from both sides.
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In previous example, the function was undefined at x = 1, but the limit still existed.
Any combination is possible:
Function defined, Limit existsFunction undefined, Limit existsFunction defined, Limit DNEFunction undefined, Limit DNE
Times when one might have an answer and the other does not:
A) Holes
B) Boundaries of Piecewise Functions
(More?)
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Defined
Exists
Abe approaches from the left... Ulysses approaches from the right...
Function Defined; Limit Exists
The club is there, and they meet.
A reallife example of the different combinations, using two friends and a club:
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Exists
Function Undefined; Limit Exists
Abe approaches from the left... Ulysses approaches from the right...
The club exploded, but they approach the leftover hole.
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DNE
Defined
Function Defined; Limit DNE
The club is there, and Abe went to the club, but Ulysses got lost.
Abe approaches from the left... Ulysses approaches from the right...
?
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September 24, 2015
DNE
Function Undefined; Limit DNE
The club exploded, but Abe showed up. Ulysses got lost.
Abe approaches from the left... Ulysses approaches from the right...
?
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September 24, 2015
Guidelines for evaluating a limit of a NONpiecewise function.
Plug in the value of x we are concerned with. We will have a few possible outcomes.
1. We get a real number back. This is our answer.
2. We get a nonzero divided by zero. This represents an asymptote, and the limit DNE.
3. We get 0/0. This often represents a hole. The limit might exist!
a) Factor and cancel?
b) Multiply by conjugates?
4. We get ∞/∞. This is undefined. The limit might exist!
a) Factor and cancel?
b) Multiply by conjugates?
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Evaluate the following limits:
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x
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Guidelines for evaluating a limit of a piecewise function.
If c (the value of x we are concerned with) is not at a boundary, follow the guidelines for nonpiecewise.
If c does fall at a boundary...
Make sure the function is defined immediately to the left and right of c. If not, limit DNE.
Plug in values of x immediately left and right of c (so immediate that the boundary itself works).
If the function's values approach the same number from both sides, the limit is that number.
Else: DNE
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In order for to exist, the following must be true...
This says that for the limit to exist at c, the left sided limit at c must equal the right sided limit at c.
This also allows us to evaluate OneSided Limits
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Evaluate the following limits:
where
where
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Evaluate the following limits:
Notice that the function is not defined at x = 2, but the limit still exists. It is possible for a function to have a limit at c that is different from f(c).
where
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Evaluate the following quantities:
Here, f(0) = 5, but the limit as x approaches 0 is 4.
Given:
Find: and
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Evaluate the following limit:
Here, with direct substitution yielding 0/0, we try factoring or multiplying by conjugates. Neither works, so we look at the onesided limits. Examining the graph and/or table helps.
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Examination of the graph/table shows the following:
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What about algebraically?
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These work for any nonzero constant c, and they make sense.
Operations with infinity and zero that we need to know about!!!
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Operations with infinity and zero that we need to know about!!!
The following are all undefined; they don't always make sense.
(Undefined and DNE are not the same thing. An undefined answer CAN exist, it just won't be the same thing every time!!)
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Horizontal Asymptote (Precalc Review):
y = 0: if the degree of the numerator is less than the degree of the denominator. (degree = largest exponent)
y = a/b: if the degrees are equal, where a is the leading coefficient of the numerator and b is the leadingcoefficient of the denominator.
DNE: if the degree of the numerator is larger than thedegree of the denominator.
Limits at Infinity:
See above (works for pos & neg infinity).This is a shortcut to the answer; you will be required to back it up with calculus.
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Review: Find the Horizontal Asymptote. New: Find the limit.
Limits at Infinity
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We need to JUSTIFY the answers for limits at infinity. The process is shown below. It involves factoring!!
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Properties of Limits
where
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Day 1Pg 57:157 E.O.O.
Day 2Pg 57: 359