Russ P. answered • 10/30/14
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Krisli,
End behavior of f(x) is approximated by its term with the highest power of x, or x4 here. Whether x is negative or positive, the value of x4 will always be positive because the power is even. For example,
at x=-10, x4 = (-10)4 = 10,000; f(-10) = (-11)2(-13)(-9) = 14,157; so x4 is already 70.6% of actual f(x).
at x=+100, x4 = (10)8; f(100)= 0.96020397 x (10)8 ; so x4 is now 104.14% of actual f(x).
Therefore,
(a) lim f(x) as x → -∞ or +∞ = lim f(x) as |x| → ∞ = lim x4 as |x| → ∞ = ∞.
(b) the zeroes of f(x) are shown by its given factorized form. Set each factor to zero, one by one:
(x-1)2 = 0 means (x-1) = 0 or x = +1, +1 (or twice at x=+1); for x=+1, f(1) just touches zero (local maximum)
(x-3) = 0 means x = +3 is another zero; f(x) crosses from negative to positive at this zero.
(x+1) = 0 means x = -1 is final zero; f(x) also crosses from negative to positive at this zero.
Being a fourth power polynomial, the max number of zeroes is 4.
(c) It looks like a giant vertical rounded "W", where
the most negative values f(x) = -3 occur at x=0 and x=+3, the bottom points of the "w"
the middle top point of f(x) = 0 and occurs at x = +1
the two outside lines both go to infinity, crossing f(x) = 0 from negative to positive at x=-1 and x=+3.
Remember, the 3 sharp corners of my printed "W" do not exist. f(x) rounds them out continuously.
Bob A.
I take exception to your statement:
"(a) lim f(x) as x → -∞ or +∞ = lim f(x) as |x| → ∞ = lim x4 as |x| → ∞ = ∞."
For a function to have a limit there are a number of conditions.
such as the left and right limits must both be the same.
Another it that the function must approach a BOUNDED value.
Limits cannot strictly equal infinity.
The function can approach infinity at a value (as this does at + and - infinity)
But a limit cannot BE infinity
since infinity has no limit.
Often scientist and engineers will say a limit is infinity
when they mean a function is approaching infinity.
But that is not really correct according to the rules of mathematics
and the definition of a limit.
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10/30/14
Russ P.
To me that's semantics unless we're being totally rigorous.
You choose any value of +N no matter how large, I can find +- x's in the current example, such that
f(x) and f(-x) > N. Therefore, as x approaches +-infinity so does f(x) approach positive infinity, even though infinity is not itself a number. And I can elaborate this argument further, by finding upper and lower bounding functions for f(x) which also follow this limit behavior and squeeze f(x) in between so all 3 approach infinity. But why bother?
So I think it will just confuse the student at this point.
Russ
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10/30/14
Bob A.
Yes semantics - semantics being the meaning.
A limit has to have a limit - and infinite does not have a limit.
At an x value a function may approach infinity but that is Not a Limit.
In every pre-calc class and calculus class, and in any maths text I have seen,
this is always spelled out as being one of the criteria of a function having a limit.
On a math test it would be marked wrong.
On the SAT you would get it wrong.
- Because you showed you didn't really understand what a limit was.
You can't say 2+2 = 4.00000000000000001 because it is about right;
so don't say something has a limit of infinity because it is sort of right
and because you expect the other person to "get" what you mean.
The beauty of mathematics as the universal language of science used
to describe the universe, is that it is a rigorous and exact language.
It does not have the ambiguities and vagueness and is not full of
the misunderstandings possible with natural human languages.
A sentence like "Lets see how much wood a woodchuck would chuck sitting in the downs down by the sea?" just doesn't happen.
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10/30/14
Russ P.
I agree with you in the “finite portions” of the domain and functional values. But as Captain Kirk in “Star Trek” would probably say: no matter how far you go on the real number line, there is always a “beyond”
where an infinity of arbitrarily large values reside. Mathematicians introduced the infinity symbol to represent the “beyond” recognizing that it is not any single value.
You can do arithmetic operations within the “finite portions” using specific finite values, but arithmetic doesn’t work in the “beyond” regions. Thus, infinity plus infinity is not 2 infinities but still only infinity:
“beyond” plus “beyond” (not being specific numbers) is still only “beyond”. Similarly, a billion times infinity is still only infinity. In effect, once you get into the “beyond”,
you stay in the “beyond”.
So rigorously, you are correct. In the “finite portion” a limit L is a specific number, and operationally, you can demonstrate convergence toward a specific limit making the difference between that limit L and the value of the function
f(x), | L-f(x) |, arbitrarily close to zero by finding suitable values of x as x approaches xofrom either side so there is no discontinuity in the function and a single limit exists.
But in the “beyond” you can’t demonstrate such convergence to zero since arithmetic doesn’t apply, and infinity minus infinity is still infinity. However, you can say that when values are in the “beyond”, they stay
in the “beyond”. In that sense, when talking about the “beyond” the “Limit” is the “beyond”. Else, why introduce infinity? It provides a convenient terse notation to replace
all this clumsy verbiage. And why say in the above problem, as x approaches infinity? Because we want to say tersely: as x becomes “arbitrarily large” and goes into the “beyond”.
On the other point raised. Yes, mathematical language is much more precise than natural language such as English. But it is still deficient. All of mathematics cannot be derived from a set of fundamental axioms. Hilbert and Lord Bertrand Russell tried
and failed. Then in 1931, Kurt Godel proved that Hilbert’s quest was impossible. There will always be statements about the natural numbers that are true, but cannot be proven so within the system of mathematics. So total consistency cannot be demonstrated.
Loosely speaking, there remain some residual inconsistences, ambiguities, or holes that cannot be fixed. So this is an “uncertainty principle” for mathematics.
Why not? Physics has its Heisenberg Uncertainty principle that limits simultaneous knowledge of a particle’s position and velocity: infinite accuracy in both measurements is not possible.
Russ
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10/31/14
KRISLI M.
10/30/14