Question

How do you evaluate `x / sqrt (4x^2 + 2x +1)` as x approaches negative infinity?

Answer 1

`-1/2`

Explanation:

As `x` approaches either infinity, the `2x+1` becomes increasing an monotonically smaller compared to the `4x^2`. So ignore it to find the limit: `x/sqrt(4x^2)=x/(2x)=1/2`. A more rigorous procedure would involve the delta-epsilon construction. However, in cases of simple rational fractions, or something close to them like this, all you need do, usually, is ignore all powers except the highest in the numerator and denominator.