Evaluate the Limit

Question

How do you evaluate the limit...

lim t-->inf (square root t²+at+b - (t-c))

The square root is over t²+at+b

Answer 1

lim [√(t² + at + b) - (t - c)] = lim{[√(t² + at + b) - (t - c)}√(t² + at + b) + (t - c)]/√(t² + at + b) + (t - c)]} =

t→ ∞ t→∞

= lim{(t² + at + b - t² + 2ct - c²)/[t√(1 + a/t + b/t²) + t(1 - c/t)]} = lim{[t(a + 2c + b/t - c²/t)/[t(√(1 + a/t + b/t²) + 1 -

t→∞

c/t)]} = (a + 2c)/2

Answer 2

If the portion inside the square root was in a square (such as √(x+5)²) then it would be easy to evaluate, right?

So, let's complete the square:

t² + at + b =

t² + at + (a/2)² + b - (a/2)² =

(t + a/2)² + b - (a/2)²

Now, we have

and when t approaches infinity the constant terms under the radical ("b" and "(a/2)²" become insignificant leaving us with √(t + a/2)² which is just t + a/2

Combining that with "- (t - c)" we get a/2 + c

So the limit as t approaches infinity is a/2 + c