Question

If x be nearly equal to 1, show that

Answer 1

\[\frac{ mx ^{n}-nx ^{m} }{ x ^{n}-x ^{m} }=\frac{ 1 }{ 1-x }\]

Answer 2

Prove using only a single side

Answer 3

\[x \approx 1\]
Let \[x=1+h\]

Answer 4

Where h be a small a small change , so small that h & higher power can be neglected

Answer 5

I am stuck in the ending so plz help

Answer 6

The denominator tends to zero.

Answer 7

i didn't understand

Answer 8

wait i am posting a type of this question that i did myself

Answer 9

How do you define \(x \approx 1\)?

Answer 10

wait @ParthKohli

Answer 11

x is approximately equal to 1

Answer 12

How do you define "approximate"? Do you have an epsilon?

Answer 13

the question says x is nearly equal to 1 which means approx. equal to 1 so we consider h as a very small value

Answer 14

Define "small"

Answer 15

are you from India ?

Answer 16

Do you know the geometric series?

Answer 17

i am from Pakistan studying in australia

Answer 18

\[\large
\sum_{k=1}^{n-1}x^k =\frac{1-x^n}{1-x}
\]

Answer 19

Might help, not sure.

Answer 20

can u explain the way i have done in prev. question that I've posted

Answer 21

look i am showing i got stuck

Answer 22

Also,\[\dfrac{a}{1 - r}\]is the sum of an infinite geometric sequence when \(|r|<1\)

Answer 23

I don't know if that helps either.

Answer 24

Using L.H.S
\[\frac{ m(1+h)^n - n(1+h)^m }{ (1+h)^n - (1+h)^m }\]

Answer 25

\[\frac{ m(1+nh+...)-n(1+mh+...) }{ (1+nh+...)-(1+mh+...) }\]
Neglecting terms containing high powers
\[\frac{ m(1+nh)-n(1+mh) }{ (1+nh)-(1+mh) }\]

Answer 26

\[\frac{ m+mnh-n-mnh }{ 1+nh-1-mh }\]

Answer 27

now from here what to do to prove \[\frac{ 1 }{ 1-x }\]

Answer 28

remember x=1+h

Answer 29

i got stuck at a point where i got -1/h

Answer 30

plz help

Answer 31

h=x-1


-_-

Answer 32

oh my... 0_0 why didn't this come to my mind

Answer 33

thank u very much @shubhamsrg

Answer 34

:P

Answer 35

This is quite better than the proof I was typing. hmm..

Answer 36

\[\frac{ m-n }{ nh-mh }\]

Answer 37

-1/h

Answer 38

\[-\frac{ 1 }{ 1-x }\]

Answer 39

negative sign?? :/

Answer 40

@shubhamsrg

Answer 41

h=x-1
and not 1-x

Answer 42

\[-\dfrac{1}{x - 1}\]

Answer 43

Oh, already posted. I shouldn't keep looking at my keyboard all the time >_<

Answer 44

ok ok

Answer 45

thanks to all :)