If limx →2 f(x)=-8, find limx →2[f(x)]^2 / 3.

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Key Concepts

Limit of a Function

This concept involves analyzing the value that a function approaches as its input nears a particular point. In this context, it is important to understand that if a function approaches a specific value from both sides of a point, then that value is considered the limit at that point.

Continuity and Composition of Functions

For many functions, if the inner function has a limit as x approaches a particular value and the outer function is continuous at that limit, then the limit of the composition is the same as the outer function evaluated at the limit of the inner function. This principle is key in handling limits involving operations like exponentiation.

Fractional Exponents

Fractional exponents indicate both a power and a root operation. They are often interpreted by first applying the power and then the root (or vice versa), which is important when manipulating expressions under limits. This understanding allows one to simplify complex expressions, such as those involving negative bases raised to fractional powers, by breaking the operations into manageable steps.

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