# doubt

• Oct 12th 2006, 10:50 AM
Aglaia
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..
• Oct 12th 2006, 11:19 AM
TD!
Your answer is correct. Why do you think the left limit doesn't exist?
• Oct 12th 2006, 11:31 AM
Aglaia
because the function is defined only for x>=0
• Oct 12th 2006, 11:33 AM
TD!

What's your thought on this: is f(x) = sqrt(x) continuous at x = 0?
• Oct 12th 2006, 11:40 AM
Aglaia
well the value exists at x=0 so why should it be discontinuos there?i think it is contiuous..
• Oct 12th 2006, 11:42 AM
TD!
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?
• Oct 12th 2006, 11:48 AM
Aglaia
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?
• Oct 12th 2006, 11:54 AM
TD!
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.
• Oct 12th 2006, 11:59 AM
Aglaia
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?
• Oct 12th 2006, 11:59 AM
TD!