
doubt
if i have a function defined like this:
f(x)=(1+x)^(2/x) for x>0
k for x=0
at zero 0 the function is continuous for which value of k? but, to be continuous the limit must exist and be equal to the value of the function at that point; i might say e^2 but actually the left limit does not exist...that is what makes me perplex..

Your answer is correct. Why do you think the left limit doesn't exist?

because the function is defined only for x>=0

Oh I didn't read that.
What's your thought on this: is f(x) = sqrt(x) continuous at x = 0?

well the value exists at x=0 so why should it be discontinuos there?i think it is contiuous..

Being defined at x = a isn't a sufficient condition for being continous there, but you are correct: sqrt(x) is continuous at x = 0.
However, the left limit for x approaching 0 of sqrt(x) is meaningless, since sqrt(x) is only defined for x > 0.
Do you see where I'm going?

that we have to take into consideration the domain too? because the very question of concerning about the left limit is wrong as it is out o the domain, of what interests us?

Since f isn't defined for x < 0, it is meaningless to look at the left limit for x approaching 0.
In this case, you are only to consider the right limit since f is only defined for x > 0.

ok, so the same for the previous function; i don't have to concern myself about a thing, a part that is not contemplated by the function; k=e^2 then is right, isn' t it?

That's the correct answer indeed!

you're really kind!!!you made me reason about that!thank you!


The general result of what TD! said is like this.
A function is said to be continous on a closed interval [a,b] is it is continous on (a,b) and the limit from the left at a is f(a) and the limit from the right at b is f(b).