Question

evaluate the limit using the appropriate properties of limits. (if the limit is
lim _{x
ightarrow infty}left(sqrt{\frac{64 x^{3}+9 x - 7}{2 - 6 x + x^{3}}}
ight)

Explanation:

Step1: Divide numerator and denominator by \(x^{3}\)

$$\lim_{x ightarrow\infty}\sqrt{\frac{64x^{3}+9x - 7}{2-6x+x^{3}}}=\lim_{x ightarrow\infty}\sqrt{\frac{64+\frac{9}{x^{2}}-\frac{7}{x^{3}}}{\frac{2}{x^{3}}-\frac{6}{x^{2}}+1}}$$

Step2: Apply the limit property \(\lim_{x

ightarrow\infty}\frac{1}{x^{n}} = 0\) for \(n>0\)
As \(x
ightarrow\infty\), \(\frac{9}{x^{2}}
ightarrow0\), \(\frac{7}{x^{3}}
ightarrow0\), \(\frac{2}{x^{3}}
ightarrow0\), \(\frac{6}{x^{2}}
ightarrow0\)
So, \(\lim_{x
ightarrow\infty}\sqrt{\frac{64+\frac{9}{x^{2}}-\frac{7}{x^{3}}}{\frac{2}{x^{3}}-\frac{6}{x^{2}}+1}}=\sqrt{\frac{64 + 0-0}{0 - 0+1}}\)

Step3: Simplify the expression

\(\sqrt{\frac{64}{1}}=\sqrt{64}\)

Answer:

\(8\)