Question

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$a) \lim_{x \to -\infty} x^{5} - 7x^{2} + 3 \\\\\\b) \lim_{x \to -\infty} \frac{x^{3}+2x-1 }{x^{2} -2x-1} \\\\\\c) \lim_{x \to \infty} \frac{3x^{2}+1 }{7x^{3} +1}$

Answer 1

Resposta:

$a) -\infty\\b) -\infty\\c) 0$

Se puder dar melhor resposta, agradeço.

Explicação passo-a-passo:

a) $\lim_{x \to -\infty} x^5-7x^2+3$

   $\lim_{x \to- \infty} x^5(1-\frac{7x^2}{x^5} +\frac{3}{x^5} )$

   $\lim_{x \to- \infty} (1-\frac{7x^2}{x^5} +\frac{3}{x^5} )=\lim_{x \to- \infty} 1=1$

   $\lim_{x \to -\infty} x^5$·1 = $\lim_{x \to -\infty} x^5$ =$-\infty$

   $\lim_{x \to -\infty} x^5-7x^2+3$ = $-\infty$

b)  $\lim_{x \to -\infty} \frac{x^3+2x-1}{x^2-2x-1}$

    $\lim_{x \to -\infty} \frac{x^3}{x^2} \frac{1+\frac{2x}{x^3} -\frac{1}{x^3} 1}{1-\frac{2x}{x^2} -\frac{1}{x^2} }$

    $\lim_{x \to -\infty} \frac{x^3}{x^2}$ = $\lim_{x \to -\infty} x$  = $-\infty$

     $\lim_{x \to -\infty} \frac{x^3+2x-1}{x^2-2x-1}$ = $-\infty$

c)    $\lim_{x \to \infty} \frac{3x^2+1}{7x^3+1}$

      $\lim_{x \to \infty} \frac{x^2}{x^3} \frac{3x^2+1}{7x^3+1}$

      $\lim_{x \to \infty} \frac{x^2}{x^3}$

      $\lim_{x \to \infty} \frac{1}{x}$ = $\lim_{x \to -\infty} 0}$ = $0$

      $\lim_{x \to \infty} \frac{3x^2+1}{7x^3+1} = 0$