If $\displaystyle \lim_{x\to a}g(x)$ exists then is it proper to say that

$\displaystyle

\lim_{x\to a}f(g(x))=f(\lim_{x\to a}g(x))

$provided f(x) is continuous at x=a.

I read this somewhere and do not believe it to be authentic.

I have come across many examples which appear to support the above statement but then there are other examples which do not support this.

If it is true then is there a rigorous proof in support of this.

One example that I have is as follows:

$\displaystyle f(x)=[x]$

$\displaystyle

g(x)=\frac{sinx}{x}

$

In this case is

$\displaystyle

\lim_{x\to 0}f(g(x))=f(\lim_{x\to 0}g(x))

$

Another example that i have is:

$\displaystyle f(x)=[x]$

$\displaystyle

g(x)=\frac{tanx}{x}

$

In this case is

$\displaystyle

\lim_{x\to 0}f(g(x))=f(\lim_{x\to 0}g(x))

$

Would be really grateful if someone verifies this for me

Thanx in advance

Also i must mention that [x] denotes the greatest integer less than or equal to x