If then .
Hello, I read the stickied delta epsilon guide at the top of the page and found it very helpful for finding an epsilon for a given delta. The guide represents delta = epsilon/M. If M has an x in it, we need to set a bound on delta in order to get an M that is a number without any x's in it.
My problem is, for example, proving that the limit of x^3 as x approaches 3 is 27. Here is what I have so far:
In the stickied article, it says that I need to set a delta (i.e, delta = 1) and then add things to my first equation so that becomes . If i do this, my M still ends up having an x in it because the only way to turn |x -3| into a quadratic equation is to multiply by x. I can solve delta epsilon proofs, but the cubic powers are throwing me off here. Any suggestions?
Thanks for the help, Plato! I didn't realize it was that simple.
I have one more question that I was wondering if someone could look over.
I am trying to prove is continuous. Can someone please make sure I'm doing this right?
Scratch work:
Proof:
Let
(ln |epsilon| should be in the exponent)
, therefore is continuous.
Ok, I can solve for delta, but I can't create an equation to substitute it into.
Now that I have my delta written in terms of epsilon and x0, I can't figure out how to substitute it into to get it
Is it ok if I start with my delta equation , make the necessary substitions into my delta equation, and end up with for my proof? As opposed to starting with and ending with epsilon? Can I start my proof with the delta equation instead of the epsilon equation?
Here's what I'm thinking of doing:
(I raised the delta equation by the power of e on both sides)