Respond with true or false to each of the following...

Respond with true or false to each of the following assertions. Be prepared to justify your answer.
$$\text { If } \lim _{x \rightarrow a} f(x)=1, \text { then } \lim _{n \rightarrow \infty}\left\{\lim _{x \rightarrow a}[f(x)]^{n}\right\}=1.$$

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Key Concepts

Limit of a Function

This concept describes how a function behaves as its input approaches a specific value. In the context of the problem, knowing that lim??? f(x) = 1 establishes that f(x) can be made arbitrarily close to 1 by taking x sufficiently close to a.

Continuity and Composition of Limits

When a function is continuous, the limit of a composed function can be calculated as the composition of the limits. Here, since exponentiation is continuous, taking the limit as x approaches a of [f(x)]? allows us to first compute lim??? f(x) and then raise the result to the power n.

Iterated Limits and Order of Operations

The problem involves taking two limits in succession: first with respect to x and then with respect to n. This order is key because the process simplifies the expression by confirming that [f(x)]? approaches 1?, which then always equals 1 regardless of the value of n.

Exponentiation Behavior

When 1 is raised to any power, the result remains 1. Therefore, once it has been established that the inner limit leads to the base 1, exponentiation does not alter this outcome even as the exponent n tends to infinity.

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