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Hey guys,
Just looking for a little guidance on this question that my friend asked me to do.
It says:
a) Suppose that lim(x⇒a)f(x) exists and is nonzero. Prove that if lim(x⇒a)g(x) does not exist, then lim(x⇒a)f(x)g(x) also does not exist.
b) Prove the same result if lim(x⇒a)f(x) = ∞.
There are two ways that I have thought to do this. One is to try and achieve a contradiction through the assumption that the limit does exist, using the epsilon/delta definition. The other is to try and prove a 'not-limit' epsilon/proof by countering the normal epsilon/delta definition. I'm not sure which of these two methods would be simpler, or if neither? Any help would be greatly appreciated.
Thanks
Hello again guys,
I have been given a problem to solve and I'm not too sure how to go about it.
The question reads:
"A function f: R --> R is said to be even if f(-x) = f(x) for all x, and odd if f(-x) = -f(x) for all x.
a) Show that any function f: R --> R can be written as f(x) = E(x) + O(x) where E is even and O is odd.
b) Prove that there is only one way of writing f in this way"
Now so far I have not even found a way to start writing the proof of either of these things. I understand part a fully in that even and odd functions make sense to me, so I know that any even power of x must be even and any odd power must be odd, so an even/odd function can only contain even/odd terms. It also makes sense that if you collect all of the even terms together and the odd terms together than you can write the expression in a.
How could I go about proving either of these things to be true?
Any help understanding the thought process behind solving these kinds of problems would be greatly appreciated!
Thanks in advance,
Darfmore
Hey again all,
I'd just like a few ideas on how to go about solving this problem that I have been given.
The problem reads:
Consider the system of linear equations where a,b,c,d,e,f,g are real numbers and a =/= 0
ax + by +cz = 0
dy + ez = 3
fy + gz = 4
It then asks to write it in augmented matrix form (which is easy), from which we need to derive a condition that guarantees that the system has:
a) A unique solution
b) Infinitely many solutions
Now I have no idea as to how to go about answering this.
I have figured that there are 3 possibilities for the solution/s:
- Infinite
- Unique
- Inconsistent (No solution)
For it to be inconsistent then I think that d and e must be proportional to f and g. Because then if you subtract the two rows from each other you're left with an equation that reads 0 = 1, therefore inconsistent. I think that this is the only way that the third option can be arrived at.
I'm not sure about the other two, though. Any suggestions about how to go about this?
Any help would be greatly appreciated.
Thanks!
k
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