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b) $\lim _{x \rightarrow-\infty} \frac{\sqrt[3]{-2 x^{9}+6 x^{5}+2}}{x^{2}-7 x-2...

Feb 8, 2024

\lim _{x \rightarrow-\infty} \frac{\sqrt[3]{-2 x^{9}+6 x^{5}+2}}{x^{2}-7 x-2}

Solution by Steps

step 1

Identify the highest power of

x

in the numerator and denominator

step 2

Divide numerator and denominator by

x^3

, the highest power in the numerator

step 3

Simplify the expression and take the limit as

x

approaches

-\infty

step 4

Recognize that the terms with

x

in the denominator approach 0 as

x

approaches

-\infty

step 5

The leading term in the numerator is

-2x^9

and in the denominator is

x^2

step 6

After simplification, the limit is dominated by the behavior of

\frac{-2^{1/3}x^3}{1}

x

approaches

-\infty

step 7

The cube root of a negative number is negative, and as

x

approaches

-\infty

-2^{1/3}x^3

approaches

-\infty

step 8

Therefore, the limit is

-\infty

Answer

-\infty

Key Concept

Limits at infinity of rational functions with radicals

Explanation

When finding the limit at infinity of a rational function with a radical, we compare the rates of growth of the numerator and the denominator. The term with the highest power of

x

in the numerator and denominator dictates the behavior of the limit. In this case, the cube root of the leading term in the numerator grows faster than the quadratic term in the denominator as

x

approaches

-\infty

, leading to a limit of

-\infty