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?

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.

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.

To evaluate the limit of the expression [25 - t] / [5 - √(t)] as t approaches 25, we can use algebraic manipulation to simplify the expression and then substitute the value t = 25.

Starting with the expression [25 - t] / [5 - √(t)], we can multiply both the numerator and denominator by the conjugate of the denominator, which is 5 + √(t). This process is known as rationalizing the denominator.

By multiplying, we get:
[25 - t] * [5 + √(t)] / [5 - √(t)] * [5 + √(t)]

Simplifying:
= [(25 - t)*(5 + √(t))] / [5^2 - (√(t))^2]
= [(25 - t)*(5 + √(t))] / (25 - t)

Now, notice that the factor (25 - t) cancels out in the numerator and denominator. This leaves us with:
5 + √(t)

Plugging in t = 25, we obtain:
5 + √(25)
= 5 + 5
= 10

Therefore, the limit of the expression [25 - t] / [5 - √(t)] as t approaches 25 is 10.