IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

sinjan.j

this is what my text book says:

as x approaches a, ie the distance I x-a I between x and a becomes smaller and smaller.

one may ask how much smaller??

the answer is as much as u please. it may be less than o.1, 0.0000000001 and so on.

In fact we may chose any positive number

.

howsoever small the distance I x-a I may be it will always be less than

.

my question is : what is the use of taking this .

can anyone give a very easily explained answer?????

jatinroxx

U wud understand its significance wen u do Physics problems...

If dere is sm physical quantity that is varying, we assume an element of size

and then integrate...

Here

is so small dat u cant argue how small it is. It is just so small that the variation of the physical quantity is not valid on this small element.

astronautameya

look, sinjan......

thr are some situations in the physical( and even chemical) world wen u really have no choice but to approximate.
in such cases, we must take the best possible and plausible approximation, and we obtain it by getting the variable(x) as close to a value, but never really fully reach that value.
e.g, while summing up a G.P with r<1 upto infinity, what we actually take is the limit of the sum!

and so, we take this small d as the given function does not appreciably vary over this interval.

rate if satisfied

sinjan.j

can u explain that more clearly jatinroxxx, please

can u give an example too

sinjan.j

it is still not getting clear astronautameya.

please give a very simple example for me to understand.

neeraj_agarwal_1990

I think thats just a concept...

We use it in finding derivatives...when we join the points having x coordinate (a with a+h) and (a with a-h) and find slope of that line....If graph has a break at that point or it has a corner....then the lines are different and so the slopes are different....and derivative doesn't exist...

I think thats one of the applications...
we can think of other applications...(like symmetric derivative..)

sankydreams

i will explain u with a very easy example
suppose u divide a number by zero, ofcourse u will get infinity
but if u divide the same no. by a no. tending to zero, that is, a no. just less than or grtr than zero , u wud obtain a certain result, that means u will get a finite no. much less than infinity or much grtr than minus infinity
there r various such cases, so to obtain a real result or a finite result u use the concept of limits
uthe concept will become clearer as u progress in the chapter
rate if this helps u
cheers!!!

neeraj_agarwal_1990

Hey sankydreams!

If u divide a number by 0...its UNDEFINED AND NOT INFINITY!

when we divide a no. by number just greater than 0...we get +INFINITY
when we divide a no. by number just smaller than 0...we get -INFINITY

Eshaan573

refer to R.D.Sharma .limits section Introduction. I Am Sure That You Will Get The Concept.

swashata4iit

Whenever a function is undefined at a point to we take limit to see the value (finite) of the function at the nearest piont of that point of discontinuity.
This can be found greatly in physics.
For example when you try to deduce the expression of potential in a gravitational field then you take a point P force at that point be
F_p = GM/x²
Now take another point Q such that [Q-P]=

=dx where

0
ie the force IN THE LIMIT remains the same.
The expressions using calculus will be like this
_[Q]

_{[ P]} F_Q= F_P=GM/x²

Therfore,
dW= F_Pdx
Now the potential or the work done may be calculated by inegrating it.

Remember that these limits or small changes are also known as ELEMENTARY CHANGES where all the external variables remains IN THE LIMIT constant.

I hope that this discussion made your concept a bit clear.
PLZ vote me if u fnd it useful (hihi)
Besides maths is fully dependent on imagination. Thus to bring the concept of calculus the concept of limit is always needed. Dont forget that
dy/dx = _[

x_]

_{[0 ]}

y/

x.

Here also [y-x]

0

swashata4iit

Hey neeraj Sanky is correct

0/0 = infinity b'cuz u can get infinite number multiplying which by 0 u will get also 0

(Any number except 0)/0 = undefined b'coz you wont get a number multiplying which by 0 u will get anything but 0

neeraj_agarwal_1990

HEY HEY HEY!!!!

DIVISION BY 0 IS NOT ALLOWED IN MATHEMATICS!!!!

any thing divided by 0 is undefined...be it 0 or 1 or 100 or anything....

sankydreams

Hey neeraj
we generally take division of any no. by zero as infinity
we use this in calculating limits all the time, how can u deny that man??????????

0/0 is indeterminate form but finite no. divided by zero is not adn is either written infinity or is given as 'not defined'

waise bhi, infinity is not a real no.

and let me tell u that in maths, even the square root of a negative no. is not defined but i m sure that u have done an entire complex chapter on that

waddya say now?

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