Limits at infinity

Question

what is the limit at infinity and negative infinity of this radical:

sqrt(t^2-4)

This is possibly the toughest problem I've encountered when attempting to find limits at infinity because I usually deal with fractions, but I'm not sure what to do here. Please show steps on how to figure this out.

I was also wondering, are answers for negative infinity and positive infinity always the same, or are there situations where the answers can be different?

Answer 1

Answer 2

try graphing it, t on the horizontal axis, y on the vertical axis

y = sqr(t^2 -4)

square both sides

y^2 = t^2 -4 (but only use the points where y non-negative, the top half)

it's a hyperbola, two branches, with no points where -4 < t < 4

as t approaches infinity, y approaches infinity in the quadrant I part, but approaches negative infinity in the quadrant IV part

same y limit as t approaches negarive infinity. y approaches infinity in the quadrant II part, and approaches negative infinity in the quadrant III part

but since you want only the top half of the two branches of the hyperbola, y approaches positive infinity only, as t approaches either negative or positive infinity.