Evaluate lim x→1 3√x − 1 / x (Hint: x−1=(3√x)^3−1^3.)

Verified step by step guidance

1. Recognize that the problem involves evaluating a limit as \( x \to 1 \) of the expression \( \frac{\sqrt[3]{x} - 1}{x - 1} \).
2. Use the hint provided: \( x - 1 = (\sqrt[3]{x})^3 - 1^3 \). This suggests using the difference of cubes formula: \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).
3. Set \( a = \sqrt[3]{x} \) and \( b = 1 \). Then, \( a^3 - b^3 = (\sqrt[3]{x})^3 - 1^3 = x - 1 \).
4. Rewrite the expression using the difference of cubes: \( x - 1 = (\sqrt[3]{x} - 1)(\sqrt[3]{x}^2 + \sqrt[3]{x} \cdot 1 + 1^2) \).
5. Substitute this back into the limit expression: \( \frac{\sqrt[3]{x} - 1}{x - 1} = \frac{\sqrt[3]{x} - 1}{(\sqrt[3]{x} - 1)(\sqrt[3]{x}^2 + \sqrt[3]{x} + 1)} \), and simplify to evaluate the limit.

Key Concepts

Limits

In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's value at points where it may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

Factoring and Rationalization

Factoring involves rewriting an expression as a product of its factors, which can simplify the evaluation of limits. In this case, the hint suggests using the difference of cubes formula to factor the expression. Rationalization is a technique used to eliminate roots from the denominator, making it easier to compute limits.

The Difference of Cubes Formula

The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). This formula is useful for simplifying expressions where one term is a cube, as it allows us to factor the expression and potentially cancel terms, facilitating the evaluation of limits.