How do you find the limit of this function. When limit approaches 0?

Since the limit as it stands approaches 0/0, which is inherently meaningless, you need to either use L'Hopital's rule or find an algebraic way of simplifying it.

i'm assuming you probably don't yet know L'Hopital's rule (at a glance, that looks like it might get messy anyway) so algebra is the way to go.

Whenever you have a difference or a sum involving square roots and you're trying to simplify, a good thing to try is multiplying by the conjugate (if you're unfamiliar with that term, the conjugate of a - b is simply a + b, and vice versa). This is helpful because (a + b)(a - b) = a^2 - b^2, which will eliminate those square roots.

In this case, multiply top and bottom by [sqrt(1 + t) + sqrt(1 - t)]

so the numerator becomes: (1 + t) - (1 - t) = 1 + t - 1 + t

which simplifies to: 2t

The denominator is now: t[sqrt(1 + t) + sqrt(1 - t)]

The t in the numerator and the denominator will now cancel, leaving:

2/[sqrt(1 + t) + sqrt(1 - t)]

As t --> 0, the denominator goes to (sqrt1 + sqrt1) = 2

Thus we have 2/2 = 1.