Question

Evaluate the limit at t --> 25: [25 - t] / [5 - √(t)]

I multiplied it by √(x + 3) to get rid of the square root on the bottom. However, once I'm done distributing it out, I'm not sure I'm doing it right.

I get [x√(x+3) + 2x - 2] / [x + 5]

Once I plug in 25 for t, I get really big numbers and I end up getting an off number that is not correct (like 25√28 + 48 / 30)

Can someone please help?

Answer 1

Factor (25-t) into (5-sqrt(t)) and (5+sqrt(t))

Cancel top and bottom (5-sqrt(t) and left with lim t->25 of (5+sqrt(t))

which is 10.

Answer 2

The idea of rationalizing the denominator is excellent.
You may note that the denominator is of the form (A-B), so if you multiply top and bottom by (A+B), you will get (A²-B²) in the denominator.
Since you are taking the limit, t does not equal 25, so you are allowed to divide top and bottom by (25-t). Simplify and get a neat answer in no time.