Problem 36 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the limits. (a) \(\lim _{x \rightarrow \infty}\left(\frac{3}{5}\right)^{-x}\) (b) \(\lim _{x \rightarrow-\infty}\left(\frac{3}{5}\right)^{-x}\)

Step-by-Step Solution

Answer

The value of the limit as \(x\) approaches \(\infty\) for the expression \(\left(\frac{3}{5}\right)^{-x}\) is \(\infty\) and the value of the limit as \(x\) approaches \(-\infty\) for the expression \(\left(\frac{3}{5}\right)^{-x}\) is \(0\).

Step 1: Evaluate the limit at positive infinity

Firstly, let's solve the limit as \(x\) approaches \(\infty\) for the given expression \(\left(\frac{3}{5}\right)^{-x}\). The exponent is \(x\) but it is being negated, we can rewrite the expression as \(\left(\frac{5}{3}\right)^{x}\). By properties of limits, as \(x\) approaches \(\infty\), the function \(\left(\frac{5}{3}\right)^{x}\) also approaches \(\infty\). Therefore, \(\lim _{x \rightarrow \infty}\left(\frac{3}{5}\right)^{-x}\) = \(\infty\).

Step 2: Evaluate the limit at negative infinity

Next, let's solve the limit as \(x\) approaches \(-\infty\) for the expression \(\left(\frac{3}{5}\right)^{-x}\). The expression can be rewritten as \(\left(\frac{5}{3}\right)^{-x}\) or \(\left(\frac{3}{5}\right)^{x}\). Using properties of limits, as \(x\) approaches \(-\infty\), the function \(\left(\frac{3}{5}\right)^{x}\) approaches \(0\). Therefore, \(\lim _{x \rightarrow -\infty}\left(\frac{3}{5}\right)^{-x}\) = \(0\).

Key Concepts

Evaluating Limits

When we talk about evaluating limits, especially in calculus, we're dealing with the behavior of a function as the input approaches a certain value. This concept doesn't just apply to finite values; it also extends to the concept of infinity, which brings about interesting outcomes in functions. In the case of exponential functions, like the one in the exercise \( \left(\frac{3}{5}\right)^{-x} \), evaluating limits at infinity helps us understand how the function behaves as the input grows larger and larger or becomes more negative.

To evaluate these limits, we often have to manipulate the given expression to make it easier to see where the function is heading. The rewriting step in the solution, where the base is inverted and the exponent's sign changed from negative to positive, is one such simplifying transformation. It paves the way for applying the properties of limits to find the function's behavior.

Exponential Functions

Exponential functions are a type of mathematical function that involve raising a constant base to a variable exponent. In our example, we're dealing with \( \left(\frac{3}{5}\right)^{-x} \), where the base is \( \frac{3}{5} \) and \( -x \) is the exponent. An interesting characteristic of exponential functions is their rate of change: these functions grow or decay at a rate proportional to their current value, which becomes evident when looking at their behavior at infinity.

Growth and Decay

In the context of limits at infinity, an exponential function with a base greater than 1 will grow without bound, while one with a base less than 1 will decay towards zero. This foundational understanding is crucial when predicting the behavior of such functions in calculus assignments and interpreting the impact of negative exponents.

Infinity in Calculus

Infinity is a concept in calculus that often confuses students, but it's essential for describing unbounded behavior. While not a number we can quantify, it represents the idea of things that are limitless or grow without end. We see this concept when evaluating limits of functions as they approach either positive or negative infinity.

In the exercise provided, we see two differing behaviors as \( x \) approaches \( \infty \) and \( -\infty \) due to the properties of the exponential function involved. Infinity serves as a tool to express the limitless growth of the exponential function \( \left(\frac{5}{3}\right)^{x} \) as \( x \) increases indefinitely, while the function's approach to zero as \( x \) becomes more negative is reflective of the function's decay. Understanding how functions behave near the bounds of infinity is pivotal in mastering calculus concepts.

Properties of Limits

Properties of limits provide the rules and tools mathematicians use to evaluate the behavior of functions as inputs approach certain points, including infinity. These properties simplify complex expressions and allow for direct substitution, factoring, and cancellation, which are part of the arsenal for solving limit problems.

Key properties that are relevant to the exercise at hand include the limit of a constant, the limit of a power function, and the limit of an exponential function. By understanding and applying these properties, we can evaluate the limits of varied expressions with confidence. For instance, we know that for a constant \( a > 1 \) and variable \( x \) approaching infinity, the limit of \( a^x \) is infinity. Conversely, for \( 0 < a < 1 \) the limit of \( a^x \) as \( x \) approaches negative infinity is zero. These principles were directly applied to analyze and successfully solve the given exercise.