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→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

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Identify the highest power of x in the denominator. In this case, the highest power is the fifth root of x, which can be expressed as x^(1/5).

Divide both the numerator and the denominator by x^(1/5). This will help simplify the expression and make it easier to evaluate the limit.

Rewrite the expression: ((³√x)/x^(1/5) - (⁵√x)/x^(1/5)) / ((³√x)/x^(1/5) + (⁵√x)/x^(1/5)).

Simplify each term: ³√x is x^(1/3), so (³√x)/x^(1/5) becomes x^(1/3 - 1/5). Similarly, (⁵√x)/x^(1/5) becomes x^(1/5 - 1/5) = x^0 = 1.

Evaluate the limit as x approaches -∞. Consider the behavior of x^(1/3 - 1/5) as x approaches -∞, and use this to determine the limit of the entire expression.

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Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the variable approaches positive or negative infinity. This concept helps determine the end behavior of functions, particularly rational functions, by analyzing the leading terms. Understanding limits at infinity is crucial for identifying horizontal asymptotes and the overall growth or decay of functions.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. When evaluating limits of rational functions as x approaches infinity, it's essential to consider the degrees of the numerator and denominator. Simplifying the expression by dividing by the highest power of x in the denominator can reveal the function's behavior at infinity, aiding in limit calculation.

Roots and Exponents

Understanding roots and exponents is vital when dealing with expressions involving noninteger or negative powers of x. The cube root (³√x) and fifth root (⁵√x) are examples of fractional exponents, which can be rewritten as x^(1/3) and x^(1/5), respectively. Recognizing how these terms behave as x approaches infinity or negative infinity is key to simplifying and evaluating limits.