Limits and Infinity


Find the limits in Exercises 37–46.


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

Verified step by step guidance

1. 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).
2. 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))).
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)).
4. 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.
5. 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.

Key Concepts

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.