Consider the following expression: lim_{x -> infty} (sqrt(4x^2 + x) - 2x) The value of the above expression (rounded to 2 decimal places) is ___________.

GATE 2021 · Engineering Mathematics · Limits · medium

Answer: 0.25

1. Multiply by conjugate: Multiply numerator and denominator by (sqrt(4x^2+x) + 2x). Numerator: (4x^2+x) - (2x)^2 = 4x^2+x - 4x^2 = x.
2. Divide by x and take limit: Divide numerator and denominator by x: 1/(sqrt(4x^2+x)/x + 2) = 1/(sqrt(4+1/x) + 2). As x->infty, 1/x->0, giving 1/(sqrt(4)+2) = 1/(2+2) = 1/4 = 0.25.