Determine the following limits at infinity.

lim x→∞ (3+10/x^2)

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Identify the dominant term in the expression as \(x\) approaches infinity. In this case, the dominant term is the constant 3, since \(\frac{10}{x^2}\) becomes negligible as \(x\) increases.

Rewrite the expression to highlight the behavior of the terms: \(3 + \frac{10}{x^2}\).

As \(x\) approaches infinity, the term \(\frac{10}{x^2}\) approaches 0 because the denominator grows much faster than the numerator.

Recognize that the limit of a constant plus a term that approaches zero is simply the constant itself.

Conclude that the limit of the expression as \(x\) approaches infinity is the constant term, which is 3.

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Limits at Infinity

Limits at infinity refer to the behavior of a function as the input approaches infinity. This concept is crucial in calculus for understanding how functions behave in extreme cases, particularly for rational functions, polynomials, and other expressions. Evaluating limits at infinity helps determine horizontal asymptotes and the end behavior of functions.

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A rational function is a function that can be expressed as the ratio of two polynomials. When analyzing limits at infinity for rational functions, the degrees of the numerator and denominator play a significant role in determining the limit. For instance, if the degree of the numerator is less than that of the denominator, the limit approaches zero as x approaches infinity.

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In the context of limits, dominant terms are the terms in a polynomial or rational function that have the greatest impact on the function's value as x approaches infinity. For example, in the expression (3 + 10/x^2), as x becomes very large, the term 10/x^2 approaches zero, making the constant term 3 the dominant term. This concept is essential for simplifying expressions when calculating limits.

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