find the limit. (if the limit is infinite, enter \\( \\infty \\) or \\( -\\infty \\), as appropriar\n\\(…

find the limit. (if the limit is infinite, enter \\( \\infty \\) or \\( -\\infty \\), as appropriar\n\\( \\lim _ { x \\rightarrow - \\infty } \\frac { \\sqrt { 1 + 16 x ^ { 6 } } } { 5 - x ^ { 3 } } \\)

find the limit. (if the limit is infinite, enter \\( \\infty \\) or \\( -\\infty \\), as appropriar\n\\( \\lim _ { x \\rightarrow - \\infty } \\frac { \\sqrt { 1 + 16 x ^ { 6 } } } { 5 - x ^ { 3 } } \\)

Answer

Explanation:

Step1: Analyze the highest - degree terms

When (x\to-\infty), for the numerator (\sqrt{1 + 16x^{6}}), since (x^{6}\geq0) for all real (x), and when (|x|) is large, (\sqrt{1+16x^{6}}\approx\sqrt{16x^{6}}). Because (x\to-\infty), (\sqrt{16x^{6}}=- 4x^{3}) (since (\sqrt{x^{6}}=|x^{3}|=-x^{3}) when (x\lt0)). The denominator (5 - x^{3}\approx - x^{3}) when (x\to-\infty).

Step2: Calculate the limit

[ \begin{align*} \lim_{x\to-\infty}\frac{\sqrt{1 + 16x^{6}}}{5 - x^{3}}&=\lim_{x\to-\infty}\frac{-4x^{3}}{-x^{3}}\ &=\lim_{x\to-\infty}\frac{-4x^{3}}{-x^{3}}\ & = 4 \end{align*} ]

Answer:

(4)