Question

lim x->infinity 5x^2+8x-3/3x^2+2

Answer 1

Since the degree of numerator and denominator is same(2), the limit equals the ratio of leading coefficients : $\dfrac{5}{3}$

Answer 2

$Let\ v=\frac{1}{x},\ \ as\ x \to\ \infty,\ v \to\ 0.\\\\ \lim_{x \to \infty} \frac{5x^2 + 8 x - 3}{3x^2+2}\\\\= \lim_{v \to 0} \frac{\frac{5x^2+8x-3}{x^2}}{\frac{3x^2+2}{x^2}}\\\\=\lim_{v \to 0} \frac{5+\frac{8}{x}-\frac{3}{x^2}}{3+\frac{2}{x^2}}\\\\=\lim_{v \to 0} \frac{5+8v-3v^2}{3+2v^2}\\\\=\lim_{v \to 0} \frac{5+8*0-3*0}{3+2*0}\\\\=\frac{5}{3}$