Question

Evaluate the given limit. $$ \lim _{x \rightarrow \pi}\left(\frac{x-3}{x-5}\right)^{7} $$

Short answer

The limit is a constant \( \left(\frac{\pi - 3}{\pi - 5}\right)^7 \).

Step-by-step solution

Identify the Limit Structure

Here, we have a limit problem for the function \( \left(\frac{x-3}{x-5}\right)^{7} \). The limit of interest is as \( x \) approaches \( \pi \), a constant.

Substitute the Limit Value

Substitute \( x = \pi \) directly into the expression \( \frac{x-3}{x-5} \) to get: \[ \frac{\pi - 3}{\pi - 5}. \]

Evaluate the Fraction

Calculate \( \frac{\pi - 3}{\pi - 5} \). This simplifies the expression to a numeric value which we can calculate using \( \pi \).

Apply the Exponent

Once the fraction is evaluated, raise the result obtained from Step 3 to the 7th power as per the original problem, i.e., \( \left(\frac{\pi - 3}{\pi - 5}\right)^{7} \).

Compute the Limit

By computing \( \left(\frac{\pi - 3}{\pi - 5}\right)^{7} \), since there's no indeterminate form, this is the limit as \( x \) approaches \( \pi \).

Key concepts

Limit Evaluation

Evaluating limits is a fundamental concept in calculus, and it helps us understand the behavior of functions as they approach specific points. In the context of the problem statement, we are interested in knowing how the function \( \left(\frac{x-3}{x-5}\right)^{7} \) behaves as \( x \) gets closer and closer to \( \pi \). To evaluate this limit, we will use appropriate methods to determine the output of the function at or near the specified point. Typically, limit evaluation helps us analyze situations where functions might behave unpredictably or reach values we cannot directly calculate. This insight can be used in various mathematical fields and real-world applications, like calculating rates of change and predicting trends.

Substitution Method

Using the substitution method is one of the simplest ways to evaluate limits when dealing with continuous functions or those without undefined points at the value of interest. In this case, we substitute \( x = \pi \) directly into the expression \( \frac{x-3}{x-5} \). By doing this, we directly replace every instance of \( x \) in the function with \( \pi \). After substitution, the resulting expression is \( \frac{\pi - 3}{\pi - 5} \), which is a straightforward fraction that we can simplify further. This method is effective because it typically provides a quick answer to the limit problem when no complications like zero denominator or indefinite forms occur.

Limit at a Constant

The concept of evaluating a limit at a constant involves examining how a function behaves as the variable approaches a specific fixed value. Here, as \( x \rightarrow \pi \), we are focused on determining the precise value of the function at the point \( x = \pi \). This limit at a constant does not involve any indeterminate forms or infinite values; instead, it directly involves substituting the constant into the function. By assessing functions at constants, we gain precise control over understanding their local behavior, which can be crucial in mathematical modeling and problem-solving scenarios.

Fraction Simplification

Fraction simplification is an important skill needed to work through many calculus problems efficiently. After substituting \( x = \pi \) in the expression \( \frac{x-3}{x-5} \), we end up with a fraction that can be evaluated: \( \frac{\pi - 3}{\pi - 5} \). Simplifying this fraction involves substituting the numerical value of \( \pi \) (approximately 3.14159) and then performing arithmetic operations to reach a simplified numerical fraction. Once simplified, we can then raise the result to the 7th power, adhering to the original function's requirement. This step reduces the complexity of the limit problem, allowing for clear and definitive evaluation.