Question

lim f(x) = f(x0)
x->x0

equivalently
lim x->x0 [ f(x) - f(x0) ] = 0

how ?

Answer 1

hey plz help me ..

Answer 2

f(x0) doesn't depend on x, right? It's just a number. So you can take that out of the limit:
lim x->x0 [ f(x) - f(x0) ] = 0
lim x->x0 [ f(x)] - f(x0) = 0
lim x->x0 f(x) = f(x0)

Answer 3

Alternatively you can look to the definition of the limit, it will follow immediately from that.

Answer 4

are you trying again .. thanks a lot.. plz i need a little more explanation if any :(

Answer 5

Do you know what a limit is?

Answer 6

yah... i know ...... :)

Answer 7

Do you know the formal definition of a limit? With epsilon and delta?

Answer 8

yah... ofcourse.. i remember if i jst look at it again.....

Answer 9

lim x-> x0 f(x)=a means:

For all epsilon>0 theres a delta>0 such that for all x with |x-x0|<delta holds |f(x)-a|<epsilon

right?

Answer 10

yah... 100% right no doubt no mistake .. :)

Answer 11

If we take a =f(x0) we get the first limit and in the definition we have |f(x)-f(x0)|<epsilon

if we take f(x)=f(x)-f(x0) and a=0 we get the second limit and in the definition we get |f(x)-f(x0)-0|<epsilon

So if write down the definition there's no difference between the two.

Answer 12

ohhhhhhhhhhhhhhhh... i got that.. u r genius... :) :) thanks alot .. :) :) :) ..

Answer 13

i will be back with a new problem .. thanks a lot see u thomas :) :) :)

Answer 14

You're welcome, glad to help.

Answer 15

thnaks :)