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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. In this context, we analyze how the function behaves as x approaches negative infinity, focusing on the leading terms of the polynomial in both the numerator and denominator, which dominate the behavior of the function.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the given limit, the highest degree terms in the numerator and denominator are crucial for determining the limit's value, as they dictate the function's growth rate as x approaches infinity.

Rational Functions and Simplification

Rational functions are ratios of polynomial functions. To evaluate limits involving rational functions, it is often useful to simplify the expression by dividing the numerator and denominator by the highest power of x present. This simplification helps in identifying the limit more easily, especially when dealing with infinity.

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