Question

How can this limit exist?

Why does `lim_(x->0)(sqrt(x))=0`

If you plug in values on the left side of 0 they will be negative, won't they. Then the value would be non real. I really can't see how the limit of this exist.

Answer 1

Be careful about the definition of limit.

Explanation:

If we are working only in the real numbers.

Pay attention to the definition of limit of `f` at `a` being used in context.

Some definitions begin with some version of "Let `f` be defined on an open interval containing `a`, except possibly at `a` itself . . . ". In this case, `lim_(xrarr0)sqrtx` is not defined.

Some definitions are more convoluted to include cases like the one you are asking about.
We could, for example, add: If `a` is an endpoint of the domain of `f`, then `lim_(xrarra)f(x)` is to be interpreted to meant the one-sided limt from the side on which `f` is defined.