Infinite Limits

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

b. lim x→0⁻ 2 / (3x¹/³)

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Identify the type of limit: This is a one-sided limit as x approaches 0 from the left (x → 0⁻).

Analyze the expression: The function is \( \frac{2}{3x^{1/3}} \). The cube root function \( x^{1/3} \) is defined for all real numbers, including negative values.

Consider the behavior of \( x^{1/3} \) as x approaches 0 from the left: As x approaches 0 from the negative side, \( x^{1/3} \) approaches 0 from the negative side as well.

Evaluate the limit: Since the denominator \( 3x^{1/3} \) approaches 0 from the negative side, the fraction \( \frac{2}{3x^{1/3}} \) will approach negative infinity.

Conclude the limit: Therefore, the limit is \( -\infty \).

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주요 개념

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. Limits can be finite or infinite, and they are essential for defining derivatives and integrals.

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. This can result in the limit being expressed as ∞ or −∞. Understanding infinite limits is important for analyzing the behavior of functions near vertical asymptotes or points where the function tends to grow indefinitely.

Infinite LimitsFind the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.b. lim x→0⁻ 2 / (3x¹/³)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Limits

One-Sided Limits

Infinite Limits

Infinite Limits

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

b. lim x→0⁻ 2 / (3x¹/³)