Limit of \( \displaystyle \frac{4 x^{2} - 1}{2 x^{2} + x} \) as \( x \to \infty \)

Problem 1.22 · medium

1. \[ \lim_{x \to \infty}\left(\frac{4 x^{2} - 1}{2 x^{2} + x}\right) \]
   limit algebraStart with the original limit expression. Divide both numerator and denominator by x**2.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{4 - \frac{1}{x^{2}}}{2 + \frac{1}{x}}\right) \]
   algebraSimplify the fractions inside the limit.✓ Proved
3. \[ = \lim_{x \to \infty}\left(4 - \frac{1}{x^{2}}\right) \left(\lim_{x \to \infty}\left(2 + \frac{1}{x}\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
4. \[ = 2 \]
   limit simplifyEvaluate the limits of the numerator and denominator separately. Final simplification.✓ Proved