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A cube root function involves finding the number that, when multiplied by itself three times, gives the original number. In mathematical terms, the cube root of a number \( x \) is represented as \( x^{1/3} \), indicating the cube root operation. This function is particularly interesting because unlike square roots, cube roots can produce both positive and negative results depending on the sign of \( x \).

- For positive \( x \), the cube root is positive. For example, \( 8^{1/3} = 2 \).- For negative \( x \), the cube root remains negative, such as with \( (-8)^{1/3} = -2 \).

This ability to handle negative inputs while remaining functionally correct is crucial for understanding behavior as \( x \) approaches zero from either side, as it allows us to evaluate both right-hand limits and left-hand limits effectively.

The right-hand limit describes how a function behaves as \( x \) approaches a certain value from the right, or the positive side. In the context of the exercise, we need to determine \( \lim_{x \to 0^+} \frac{2}{3x^{1/3}} \). This means we're interested in what happens to this expression as \( x \) gets closer to zero from values slightly greater than zero.

As \( x \to 0^+ \):

• The expression \( x^{1/3} \) becomes very small and positive, as it's the cube root of a small positive number.
• Consequently, \( \frac{2}{3x^{1/3}} \) becomes very large, as we're dividing by a very small positive number, causing the overall value to trend towards positive infinity.

Hence, \( \lim_{x \to 0^+} \frac{2}{3x^{1/3}} = +\infty \). This shows the behavior becomes unbounded as it approaches positive zero.

The left-hand limit examines how a function behaves as \( x \) approaches a certain value from the left, or the negative side. In our exercise, we seek \( \lim_{x \to 0^-} \frac{2}{3x^{1/3}} \). This involves exploring the function as \( x \) nears zero from slightly negative numbers.

Key elements include:

• As \( x \to 0^- \): The term \( x^{1/3} \) transforms into a small negative number since the cube root of a negative value remains negative.
• Therefore, \( \frac{2}{3x^{1/3}} \) produces a large negative result. Dividing by such a small negative value drives the expression toward negative infinity.

So, \( \lim_{x \to 0^-} \frac{2}{3x^{1/3}} = -\infty \). This demonstrates the function diving down to a large negative value as it nears zero from the left.

Infinite limits occur when a function grows larger and larger without bound as \( x \) approaches a particular value. In the exercise, both right-hand and left-hand limits as \( x \to 0 \) exhibit infinite behavior, but in opposite directions.

- For \( \lim_{x \to 0^+} \frac{2}{3x^{1/3}} \), the function increases towards positive infinity.- In contrast, \( \lim_{x \to 0^-} \frac{2}{3x^{1/3}} \) drives the function towards negative infinity.

This unbounded growth or decline signifies an infinite limit, crucial in understanding discontinuities and asymptotic behavior in functions. Infinite limits highlight instances where a function does not approach a finite value but instead continually escalates or descends in magnitude.