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