Is the Limit of a Continuous Function Equal to the Limit of its Variable?

[JG89]

If f is continuous in some neighborhood of x = a, then is the following true:

$$\lim_{x \rightarrow a} f(x) = f( \lim_{x \rightarrow a} x)$$?

 

[de_brook]

If f is continuous in some neighborhood of x = a, it is also continuous at x = a, because x = a is contained in the neighborhood. The l.h.s equals the r.h.s because of the fact that the limit as x tends to 'a' of f(x) equals 'f(a)' (because of continuity of f) and on the other hand, f of the limit of x as 'x tends to a' is obviously f(a) since lim(x) = a as x --> a

 

[poutsos.A]

If f is continuous in some neighborhood of x = a, then is the following true:

$$\lim_{x \rightarrow a} f(x) = f( \lim_{x \rightarrow a} x)$$?

This an anorthodoxe way of writting : f is continuous at x=a <====>
$$\lim_{x \rightarrow a} f(x)=f(a)$$

But i suppose is correct since $$\lim_{x\rightarrow a}x = a$$

 

[lurflurf]

If f is continuous in some neighborhood of x = a, then is the following true:

$$\lim_{x \rightarrow a} f(x) = f( \lim_{x \rightarrow a} x)$$?

That is true. This makes clear the idea of continuity.