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Did the OP not intend to start at 28!?
Why should the answer be 56? The point was to add 28, not subtract 304888344611713860501503999944.
I think it was just a typo.
Anytime! I'm happy to help. Have a nice day.
Don't worry, it happens to everybody! I think you are right, I don't think assumptions are necessary for understanding - sometimes it's easier to start learning something from scratch!
Ah yes, but not all functions are continuous or defined for all x. I'm glad we got all that cleared up, though!
I suppose so, but the problem is that we cannot find the limit algebraically. This would be some sort of implicity defined plot, and cannot be expressed as y=... So we can't exactly write an expression describing the limit of y as x tends to any value.
If you are interested in finding limits with curves that are not functions, try doing it with some parametric plots. They can be very interesting looking graphs.
To find a limit for something that is not a function would probably require you to use parametric or polar coordinates. For example, you could find the limit of x as t tends to c, and the limit of y as t tends to c, for a parametric plot. Then you would have the point (x,y) for the limit as t tends to c. And for a polar plot, you would have to limit θ to avoid repetition.
In the case of your graph, there is no limit as x tends to -3 because it takes on two different values of y there.
No problem! Glad I could help.
That is where I was having troubles with it.
Is there a way to approximate it without computer software?
The red and black curves are separate functions, so we should expect two values of y at each x where f(x) and g(x) are defined. One for each function.
I am not sure I understand the question.
If you are asking whether this fact is true wherever the f(x) and g(x) are defined, then no, that is not necessary. It works on any interval (a,b) where f(x) ≤ g(x). So maybe f(x)≤g(x) only on the interval (-5,2). This fact will still hold true.
A piecewise function isn't really a sum of functions. It is a function made up of sub-functions defined on subintervals its domain.
An example of a peicewise function is
I suppose you could create a graph with piecewise notation which does not satisfy the vertical line test. Just make the subintervals of 2 or more sub-functions overlap. An example:
You cannot evaluate the limit of y as x approaches any value between -2 and 0, because y takes on two values.
Your graph is correct in displaying the inequality. Let the red line be g(x), and the black one be f(x). f(x)≤g(x) at everywhere shown, except at x=5. So f(5)>g(5). Yet,
. I think you got it!Yes, that sounds right. A piecewise function could be a good example to explain that inequality.
For example:
If you graph these, you can see that f(x)≤g(x) everywhere except at x=0. So let c=0.
We still have f(x)≤g(x) on (a,b) and c=0, with a<c<b. But here, f(c)=3>g(c)=0. Yet,
I hope this clears everything up.
And yes, f(x), although piecewise, is still a function.
I am not sure I understand your question. What do you mean by the whole?