check it up , please ....

that condition is continuity. This follows from Heine's definition of continuity

could somebody elaborate???

@prophet . I'm getting a bit confused da.

http://en.wikipedia.org/wiki/Continuous_function#Heine_definition_of_continuity

At wiki it says for fog , f is continuous if g is tending to a limit.

but here g does tend to a limit but f isn't continouus no???

koi help karo . maybe I'm not understanding it properly( what's given at wiki) . madad karo koi . check out the link.

@srinath,,, its clearly given in fiitjee's RSM dat:

let lim x->a f(x)=m and lim x->a g(x)=n

lim x->a f(g(x)) = f(lim x->a g(x)) = f(n)

if and only if f(x) is continuos at x=n.

did u get it?

the function f(x) has to be continuous at the limiting value of g(x->a)

for the ex u gave,

g(x)= x³ , f(x)=[x].

for limx->a g(x)=a³
but at x=a³, f(x) is not continuous... (provided 'a' is integre)

so the formula is not valid...

Firstly, let's see what is the definition of limit as given by Heine

Lim x→ a F(x) exists iff for every sequence x₁,x₂,...,x_n,... (we can write {x_n} for short) converging to a, the sequence {F(x_n} converges to a limit l.

If F(x) is continuous this further means that the sequence {F(x_n)} will converge to F(a)

So, now let us look at the question Lim x→ a f(g(x))

So first we must look at all sequences {x_n)} converging to a.

This in turn gives rise to a subsequence,{g(x_n)}

If the given limit exists, this subsequence will converge to some limit l

Now further if f(x) is continuous, we must have

that the {f(g(x_n))} will converge to f(l)

i.e. to f(Lim x→ a g(x))

Hence, iff f(x) is continuous and Lim x→ a g(x) exists, then Lim x→ a f(g(x)) = f(Lim x→ a g(x))

Srinath, if you are not still satisfied, please do post your doubt. I was not online much yday, so I couldnt reply in time