Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 4 / x²/⁵

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First, identify the form of the limit as x approaches 0. The expression given is \( \frac{4}{x^{2/5}} \).

Recognize that as x approaches 0, \( x^{2/5} \) also approaches 0. Since \( x^{2/5} \) is in the denominator, the expression \( \frac{4}{x^{2/5}} \) will tend towards infinity.

Consider the direction from which x approaches 0. If x approaches 0 from the positive side (x → 0⁺), \( x^{2/5} \) is positive, and thus \( \frac{4}{x^{2/5}} \) approaches positive infinity.

If x approaches 0 from the negative side (x → 0⁻), \( x^{2/5} \) is still positive because the fifth root of a negative number is negative, but squaring it makes it positive. Therefore, \( \frac{4}{x^{2/5}} \) also approaches positive infinity.

Conclude that the limit of \( \frac{4}{x^{2/5}} \) as x approaches 0 is positive infinity, regardless of the direction from which x approaches 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. In this context, as x approaches 0, the function 4/x²/⁵ may tend towards infinity or negative infinity, depending on the behavior of the denominator, which becomes very small, causing the overall expression to grow very large.

Behavior of Rational Functions

Rational functions are expressions involving ratios of polynomials. The behavior of these functions near points where the denominator approaches zero is crucial for determining limits. As x approaches 0 in 4/x²/⁵, the denominator x²/⁵ approaches zero, leading to a potential infinite limit, as the numerator remains constant and the denominator shrinks.

Power Functions and Exponents

Understanding power functions and their exponents is essential for analyzing limits involving expressions like x²/⁵. The exponent determines how rapidly the function approaches zero or infinity as x approaches a specific value. In this case, x²/⁵ indicates a root, which affects the rate at which the denominator approaches zero, influencing the limit's behavior.

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