Limits and Infinity

Find the limits in Exercises 37–46.

x²/³ + x⁻¹
lim --------------------
x→∞ x²/³ + cos²x

검증된 단계별 안내

First, identify the dominant terms in the numerator and the denominator as x approaches infinity. In this case, the dominant term in both the numerator and the denominator is x^(2/3).

Rewrite the expression by factoring out x^(2/3) from both the numerator and the denominator. This will help simplify the limit. The expression becomes: (x^(2/3) * (1 + x^(-5/3))) / (x^(2/3) * (1 + cos^2(x)/x^(2/3))).

Cancel out the common factor of x^(2/3) from the numerator and the denominator. This simplifies the expression to: (1 + x^(-5/3)) / (1 + cos^2(x)/x^(2/3)).

Evaluate the limit of each term as x approaches infinity. The term x^(-5/3) approaches 0, and cos^2(x)/x^(2/3) also approaches 0 because cos^2(x) is bounded between 0 and 1.

Substitute these limits into the simplified expression: (1 + 0) / (1 + 0) = 1. Therefore, the limit of the original expression as x approaches infinity is 1.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, we are interested in the limit of a function as x approaches infinity, which helps determine the behavior of the function at extreme values.

Dominant Terms

In limit problems, especially as x approaches infinity, the dominant term in a polynomial or rational function significantly influences the limit's value. For example, in the expression x²/³ + x⁻¹, as x becomes very large, the term x²/³ will dominate over x⁻¹, allowing us to simplify the limit calculation by focusing on the leading term.

Trigonometric Functions and Limits

Trigonometric functions, such as cos²x, oscillate between fixed values, which can affect the limit of a function. In this case, as x approaches infinity, cos²x remains bounded between 0 and 1, meaning its contribution to the limit can be considered negligible compared to polynomial terms. Understanding how these functions behave at infinity is crucial for evaluating limits involving them.

Limits and InfinityFind the limits in Exercises 37–46.x²/³ + x⁻¹lim --------------------x→∞ x²/³ + cos²x