Determine the following limits.

lim x→−∞ 40x^4+x^2+5x / √64x^8+x^6

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Identify the dominant terms in the numerator and the denominator. In the numerator, the dominant term is \$40x^4$. In the denominator, the dominant term is $\sqrt{64x^8}$.

Simplify the expression by dividing both the numerator and the denominator by the highest power of $x$ present in the dominant terms. Here, divide by $x^4$ in the numerator and $x^4$ in the denominator (since $\sqrt{64x^8} = 8x^4$).

Rewrite the expression: $\frac{40x^4 + x^2 + 5x}{\sqrt{64x^8 + x^6}} = \frac{40 + \frac{1}{x^2} + \frac{5}{x^3}}{8\sqrt{1 + \frac{1}{64x^2}}}$.

Evaluate the limit as $x \to -\infty$. As $x$ approaches $-\infty$, the terms $\frac{1}{x^2}$, $\frac{5}{x^3}$, and $\frac{1}{64x^2}$ approach 0.

The limit simplifies to $\frac{40}{8}$, which can be further simplified to find the final result.

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