Question

why limits is used

Answer 1

To find the value of a function as its input approaches a particular finite or infinite value.

Answer 2

Because sometimes functions are continual at certain points, and we want to know how the function looks very close to that value.

Answer 3

http://math.clarku.edu/~djoyce/ma120/whylimits.pdf

Answer 4

Precisely, probably the most common thing I've used limits for is in curve sketching, to find the continuity of a function at a very local point.

Answer 5

Calculus makes graph sketching so easy.

Answer 6

It really does ;)

Answer 7

We'll spend the next few weeks studying \limits." Naturally, the question is
\why limits?" Why not just go on to derivatives?
The answer involves the character of the course. This is not just a course about how to
use calculus, but a mathematics course about what calculus is.
The greatest minds of the of the 17th century, Newton and Leibniz, spent considerable
time not just inventing calculus, but trying to gure out what it is. Newton and Leibniz
created rules for dealing with derivatives and integrals, rules that lead to the word \calculus"
for the whole subject, but neither had a very good understanding of the basis of their theory.
Leibniz rested his calculus on the concept of innitesimals. Innitesimals were supposed to
be positive quantities less than any positive number. His theory required not just innites-
imals, but innitely many orders of innitesimals. He needed second-order innitesimals
smaller than any of the rst-order innitesimals, and third-order innitesimals, and so forth.
The foundations of Leibniz' innitesimals were not logically justied until the middle of the
20th century.
Newton founded his calculus on intuitive concepts of limits. He did not dene what limits
were, nor did he state the properties that he expected limits to have. The foundations of
Newton's limits were not logically justied until the 19th century.
What we're going to do is develop foundations of limits, the kind that Newton used only
intuitively. That is going to take some time.
How do derivatives depend on limits? Just how are derivatives supposed to depend on
limits? A derivative is supposed to be the rate of change of a function at an instant, what
we'll call \instantaneous rate of change." On the face of it, that makes no sense at all, since
an instant is a point in time, and nothing changes at a point in time. You need a time interval
for anything to change.
Leibniz got around that by taking a point in time to have length, although for him it
was an innitely short length. Newton didn't do it that way. Newton recognized that an
instantaneous rate of change could be found as a limit of rates of change over shorter and
shorter intervals. In fact, Newton wasn't the rst to use that idea since Fermat and others
before Newton had already done that. Here's how they did it.
Start with a function y = f(x). To help us understand, let's take x to be time, measured
in some convenient time unit, and let's take y = f(x) to be the distance travelled at time x,
measured in some convenient distance unit. Then the derivative is what we know as velocity.
The velocity doesn't have to be constant, but may change over time. It might slowly at rst
with a small velocity, and later quickly with a large velocity, or vice versa. We're trying
to determine how to nd the velocity f
0
(x) when we know the distance f(x). Finding the
derivative f
0
(x) when you know f(x) is called dierentiation.
Average rates of change. We can fairly easily compute the average rate of change,
that is, the average velocity, over an interval. Suppose we take the time interval [a; b] which
starts at time x = a and ends at time x = b. We can compute the distance travelled over
that interval as the dierence f(b) f(a). But the length of the time interval is b a. Tha

Answer 8

So no one can go off the limit. ;)

Answer 9

Ahhh a mathematician and a comedian! :-P

Answer 10

yeah fewscrewsmissing the very the well the said, It is helpful to find the continuity of the curve given by a functional relation....mainly to determine in the interval of neighbourhood which reveals the continuity of the curve

Answer 11

@FSM: I'm sure you have used them for more reasons. In fact, both differentiation and integration lie completely in the concept of limit. You can see that from their definitions, because they are nothing but limits.

Answer 12

Lol.

Answer 13

Oh very much. If you differentiate from first principles, you'll see the importance of limits. Of course we find shortcuts - y = x^n -> y' = nx^(n-1) etc - but it all stems from one area. I mean the explicit use of a limit for the purpose of finding a limit though.

Answer 14

Mr.Math is correct, but how many people do actually use limit in calculus?

Answer 15

Check out this books:

http://edoutreach.wpafb.af.mil/ed_outreach/pages/ed-resources/calculus_wo_limits.pdf

http://www.amazon.com/Calculus-Without-Limits-John-Sparks/dp/1418441244

Answer 16

Muggles will always find a way :P

Answer 17

Most universities, as I know, don't ask students for the proofs of derivatives and integrals formulas. But in fact, most of them are derived from the definitions (i.e limits).

Answer 18

In the last applied mathematics unit I did, we learnt how differentiation rose from limits, more or less, but it was only very briefly touched on. I only found out about first principles and how the shortcuts we make are derived from more extensive methods through my own findings.

Answer 19

If you don't understand limits you can still solve calculus problems but then you will never know the real power of calculus.