Key to "Compute the limits. Justify your answer. a) limx→∞16x3−19x2+7x+115x2−10x3−19x+100..."

geduiwelh

Answered question

2021-10-16

Compute the limits. Justify your answer.
a)

lim_{x \to \infty} \frac{16 x^{3} - 19 x^{2} + 7 x + 11}{5 x^{2} - 10 x^{3} - 19 x + 100}

lim_{x \to \infty} \frac{x^{\frac{5}{7}}}{17 x + 29}

hajavaF

Skilled2021-10-17Added 90 answers

The limit of given function is computed directly by using some results
a)

lim_{x \to \infty} \frac{16 x^{3} - 19 x^{2} + 7 x + 11}{5 x^{2} - 10 x^{3} - 19 x + 100} = lim_{x \to \infty} \frac{16 x^{3} - 19 x^{2} + 7 x + 11}{- 10 x^{3} + 6 x^{2} - 19 x + 100}

= lim_{x \to \infty} \frac{x^{3} (16 - \frac{19}{x} + \frac{7}{x^{2}} + \frac{11}{x^{3}})}{x^{3} (- 10 + \frac{5}{x} - \frac{19}{x^{2}} + \frac{100}{x^{3}})}

= lim_{x \to \infty} \frac{(16 - \frac{19}{x} + \frac{7}{x^{2}} + \frac{11}{x^{3}})}{- 10 + \frac{5}{x} - \frac{19}{x^{2}} + \frac{100}{x^{3}}}}

= \frac{16}{- 10} = \frac{- 8}{5}

lim_{x \to \infty} \frac{x^{\frac{5}{7}}}{(17 x + 29)} = lim_{x \to \infty} \frac{x (\frac{x^{\frac{5}{7}}}{x})}{x (\frac{17 x + 29}{x})} = lim_{x \to \infty} \frac{(\frac{x^{\frac{5}{7}}}{x})}{(\frac{17 x + 29}{x})}

\Rightarrow lim_{x \to \infty} \frac{(\frac{1}{x^{\frac{2}{7}}})}{(17 + \frac{29}{x})} = \frac{0}{17 + 0} = 0

\Rightarrow lim_{x \to \infty} \frac{x^{\frac{5}{7}}}{17 x + 29} = 0

Jeffrey Jordon

Expert2022-06-29Added 2605 answers

Answer is given below (on video)

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