Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→∞ (2√x + x⁻¹) / (3x − 7)

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Identify the highest power of x in the denominator, which is x in this case.

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

Rewrite the expression: (2√x/x + x⁻¹/x) / (3x/x - 7/x).

Simplify each term: (2/√x + 1/x²) / (3 - 7/x).

Evaluate the limit as x approaches infinity: As x → ∞, 2/√x → 0, 1/x² → 0, and 7/x → 0, so the limit simplifies to 0/3.

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Limits at Infinity

Limits at infinity refer to the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, particularly for rational functions where the degree of the numerator and denominator can determine the limit. Analyzing limits at infinity helps in identifying horizontal asymptotes and the end behavior of functions.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The degree of the polynomials in the numerator and denominator plays a significant role in determining the limit as x approaches infinity. Understanding the structure of rational functions allows for the simplification of limits by dividing through by the highest power of x, which reveals the dominant terms that dictate the limit's value.

Dominant Terms

In the context of limits, dominant terms are the terms in a polynomial that have the highest degree and thus have the most significant impact on the function's behavior as x approaches infinity. When evaluating limits, identifying and focusing on these terms allows for simplification of the expression, making it easier to determine the limit. This concept is essential for effectively applying the technique of dividing by the highest power of x.