Question

Qual deve ser o valor de m para que (quando x tende ao infinito) Lim 10x³+5x²-2x+1/mx³+4x²-3x+9=5

Answer 1

Olá.

$\displaystyle \lim_{x \to +\infty} \frac{10x^3+5x^2-2x+1}{mx^3+4x^2-3x+9} \\ \\ \\ \lim_{x \to +\infty} \frac{x^3 \cdot (\displaystyle 10+\frac{5x^2}{x^3}-\frac{2x}{x^3}+\frac{1}{x^3})}{x^3 \cdot (\displaystyle m+\frac{4x^2}{x^3}-\frac{3x}{x^3}+\frac{9}{x^3})} \\ \\ \\ \lim_{x \to +\infty} \frac{x^3 \cdot (\displaystyle 10+\frac{5}{x}-\frac{2}{x^2}+\frac{1}{x^3})}{x^3 \cdot (\displaystyle m+\frac{4}{x}-\frac{3}{x^2}+\frac{9}{x^3})} \\ \\ \\ \lim_{x \to +\infty} \frac{10x^3}{mx^3}$

$\displaystyle \lim_{x \to +\infty} \frac{10}{m} \\ \\ \\ \frac{10}{m}=5 \\ \\ \\ \boxed{\boxed{m=2}}$