Problem 15 - APEX Calculus

Question

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{l} \text { - } \lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 10} f(x)=1, \quad f(1)=1 / 5 \\ \text { - } \lim _{x \rightarrow 1} g(x)=0, \quad \lim _{x \rightarrow 10} g(x)=\pi, \quad g(10)=\pi \end{array} $$ $$ \lim _{x \rightarrow 1} f(x)^{g(x)} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify Given Limits

We need to evaluate the limit \( \lim_{x \to 1} f(x)^{g(x)} \). From the given information, we know that \( \lim_{x \to 1} f(x) = 2 \) and \( \lim_{x \to 1} g(x) = 0 \).

Step 2: Substitute the Limits into the Expression

Replace \( f(x) \) and \( g(x) \) with their respective limits as \( x \to 1 \). Thus, the problem becomes finding \( 2^0 \).

Step 3: Simplify and Evaluate the Limit

We know that any non-zero number raised to the power of zero is 1. Therefore, \( 2^0 = 1 \).

Step 4: Conclusion

Thus, the limit \( \lim_{x \to 1} f(x)^{g(x)} = 1 \).

Key Concepts

Limit Evaluation

To evaluate limits in calculus, we determine the value that a function approaches as the input approaches a certain point. This exercise requires evaluating the limit \( \lim_{x \to 1} f(x)^{g(x)} \). It means we want to find out what the expression becomes when \( x \) gets very close to 1.

In our given problem, we are told that \( \lim_{x \to 1} f(x) = 2 \) and \( \lim_{x \to 1} g(x) = 0 \). Evaluating the limit involves substituting these values into our expression, meaning it simplifies to \( 2^0 \).

The trick with limit evaluation is knowing what each part of the expression becomes as \( x \) approaches the limit point. Here, 2 is the base and 0 is the exponent, resulting in the classic power property where any number to the zero power is 1, i.e., \( 2^0 = 1 \). Thus, \( \lim_{x \to 1} f(x)^{g(x)} = 1 \).

Limit Properties

Understanding limit properties simplifies evaluating complex functions. Properties such as limit sum, product, and quotient are fundamental tools. Here, we specifically rely on the property regarding exponentiation at a limit.

When we compute \( 2^0 \), we apply the basic exponentiation limit property that any non-zero base raised to an exponent of zero results in one. This property is among the simplest yet vital for solving calculus problems involving limits.

Another critical concept in limit evaluation is the behavior of functions as they approach specific points, particularly when dealing with infinity or indeterminate forms. Here, though, we only utilize the limit of a function to a whole number, demonstrating one of the more straightforward applications of these properties.

Calculus Problems

Calculus problems often fall into categories, such as differentiation and integration, but limit evaluation is another key area. Solving these involves identifying function behaviors as they approach certain points. The expression \( \lim_{x \to 1} f(x)^{g(x)} \) embodies such a problem.

Breaking it down, interpreting given limits, and substituting helps navigate complexities. In this exercise, upon determining \( f(x) = 2 \) and \( g(x) = 0 \) as \( x \to 1 \), the problem reduces to straightforward arithmetic: calculating \( 2^0 \).

Approaching calculus problems like this relies on a systematic method: identify given data, apply appropriate properties, and simplify the expressions. Armed with these tactics, one's skill in unraveling the complexities of calculus problems grows substantially, revealing both the power and elegance of limit evaluations.