Example 6

Find

All the limits involved exist, so we can use the limit rules to compute the limit as follows. First we find the limit of the expression inside the radical.

Now we find the answer to the original problem.

There are three ways in which a limit lim_x→c f(x) can fail to exist:

(1)    f(x) is undefined for some x which is infinitely close but not equal to c.

(2)    f(x) is infinite for some x which is infinitely close but not equal to c.

(3)    The standard part of f(x) is different for different numbers x which are infinitely close but not equal to c.