i need precalculus limits in math to be explained thoroughly...like right now =P?

Question

i'm having urgent homework problems..grr. so anyways,

a [sub]n = (n ^3 -1 ) / (2 + n ^2 )

the limit is said to not exist (in the back of the book) but i have no idea why. my teacher didn't explain very good today; we didn't have that much time. so any help is appreciated. i may have other questions about it too, but don't worry, i won't ask for you to give me answers to my homework or anything. i'm just looking to understand this stuff.

Answer 1

Don't quote me on this, but I think that since there are no numbers in the denominator that would make it equal zero, you can just say that since the exponent in the numerator is higher, the function wouldn't approach a limit. It would just keep going on forever as if it were a line.

Answer 2

To say that "the limit does not exist" simply means that "as n becomes very large," the right member of the equation does not approach any particular number. Instead, it either grows without bound (to positive or negative infinity), or else it oscillates among two or more values and doesn't settle down.

In this example, you'll find that the right member grows without bound as n becomes very large. In fact, the right member approaches n (for the reason explained in the next two paragraphs). But this means that there is no limit on the right side, since it will continue increasing as n increases.

Here's why the right member approaches n:
When ne is very big, n^3 - 1 is very close to n^3 (i.e., the percentage difference between n^3 and n^3 - 1 becomes insignificant. Similarly, 2 + n^2 gets to be nearly the same as n^2. (The difference between n^2 and 2 + n^2 remains equal to 2, but that 2 becomes a very small percentage of 2 + n^2 as n increases.)

So for large values of n, (n^3 - 1) / (2 + n^2) is essentially equal to n^3 / n^2, which equals n. This leads to the conclusion described above.

Answer 3

In layman terms, a limit is said not to exist if the limit tends to infinity (since infinity doesn't exist).

For the question above, let us express it in a proper fraction.

(n^3-1)/(2+n^2) = [n(2+n^2)-2n]/(2+n^2)
= n - (2n)/(2+n^2)
Pay particular attention to the first n in the simplified expression.

Now try to picture this. If the value of n keeps on increasing and increasing (ie, tends towards infinity), the value of the expression will keep on increasing. Why? Because of the first n in the simplified expression. You keep on increasing the value of n, and n in the simplified expression also keeps on increasing. Therefore, the entire expression tends to infinity, so the limit doesn't exist.

Confusing, i think.
**************************
If you noticed that I left out the last part of my simplified expression, and if you aren't confused, then carry on reading.
(2n)/(2+n^2)
As n tends to infinity, and n rises slower than n^2 (n<
small
-------
large

And small numbers divided by very large numbers give a value close to zero. So (2n)/(2+n^2) tends to 0.
Go back to the entire simplifed expression again. Can you figure out why the entire expression now tends to infinity?

Answer 4

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