Determine the following limits.

lim x→∞ x^4+7 / x^5+x^2−x

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Identify the highest power of x in the numerator and the denominator. In this case, the highest power in the numerator is x^4 and in the denominator is x^5.

Divide every term in the numerator and the denominator by x^5, the highest power of x in the denominator.

Rewrite the expression: \( \lim_{{x \to \infty}} \frac{x^4/x^5 + 7/x^5}{x^5/x^5 + x^2/x^5 - x/x^5} \).

Simplify each term: \( \lim_{{x \to \infty}} \frac{1/x + 7/x^5}{1 + 1/x^3 - 1/x^4} \).

Evaluate the limit as x approaches infinity. As x becomes very large, terms with x in the denominator approach zero.

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주요 개념

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Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the variable approaches infinity. In this context, we analyze how the function behaves when x becomes very large, which often simplifies the expression by focusing on the highest degree terms in the numerator and denominator.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Understanding the degrees of the polynomials in both the numerator and denominator is crucial for determining the limit, as the highest degree terms dominate the behavior of the function as x approaches infinity.

Dominant Terms

Dominant terms are the terms in a polynomial that have the highest degree and thus have the most significant impact on the function's value as x approaches infinity. In the limit calculation, we can simplify the expression by focusing only on these dominant terms, allowing us to easily determine the limit's value.

Determine the following limits. lim x→∞ x^4+7 / x^5+x^2−x