Problem 63 - Applied Calculus

Question

\(61-64\). Which of the two limits exists? a. \(\lim _{x \rightarrow-\infty} e^{3 x}\) b. \(\lim _{x \rightarrow-\infty} e^{-3 x}\)

Step-by-Step Solution

Answer

Only \( \lim_{x \rightarrow -\infty} e^{3x} \) exists; it equals 0.

Step 1: Understanding the Limits

We need to find which of the two limits exist: 1. \lim_{x \rightarrow -\infty} e^{3x}2. \lim_{x \rightarrow -\infty} e^{-3x}. Both expressions involve exponential functions with a variable in the exponent, meaning their behavior as \(x\) approaches negative infinity will primarily be influenced by the exponents.

Step 2: Analyzing the First Limit

Consider \lim_{x \rightarrow -\infty} e^{3x}.When \(x\) approaches \(-\infty\), \(3x\) becomes even more negatively large, resulting in a very small positive value for \(e^{3x}\). As exponents of \(e\) approach large negative numbers, the expression approaches 0.

Step 3: Analyzing the Second Limit

Now consider \lim_{x \rightarrow -\infty} e^{-3x}.As \(x\) approaches \(-\infty\), \(-3x\) turns into a large positive number due to the negative sign. Therefore, \(e^{-3x}\) tends towards \(+\infty\) because \(e\) raised to a very large positive number increases without bound.

Step 4: Determining which limits exist

From our analysis:- \lim_{x \rightarrow -\infty} e^{3x} = 0 exists because it approaches a finite number.- \lim_{x \rightarrow -\infty} e^{-3x} does not exist in a meaningful, finite sense, as it tends to \(+\infty\).

Key Concepts

Exponential Functions

Exponential functions are incredibly important in calculus and are often encountered in various forms. An exponential function is a mathematical expression where a constant base, often denoted by e (Euler's number, approximately 2.718), is raised to a variable exponent. This results in a function that grows or decays multiplicatively as the variable changes. For example, expressions like \( e^{3x} \) and \( e^{-3x} \) are typical exponential functions with the variable \( x \) as an exponent.

• An exponential function with a positive exponent, such as \( e^{3x} \), grows rapidly as \( x \) increases because the power of the base (e) increases.
• When the exponent is negative, like in \( e^{-3x} \), the function represents exponential decay, shrinking rapidly as \( x \) becomes more positive.

This behavior of exponential functions is crucial when analyzing how they behave as variable inputs approach certain limits. Understanding these functions lays the groundwork for exploring deeper mathematical concepts in calculus.

Infinity in Calculus

Infinity in calculus represents a concept where values grow indefinitely large (positive infinity) or indefinitely small (negative infinity). It is not a number but an idea used to describe the endless nature of growth or decay in mathematical expressions.

• When we examine an expression such as \( \lim_{x \to -\infty} e^{3x} \), we are investigating how the function behaves as \( x \) moves towards negative infinity.
• As \( x \) approaches negative infinity, an expression like \( 3x \) becomes greatly negative, leading \( e^{3x} \) to approach zero.
• Conversely, with \( \lim_{x \to -\infty} e^{-3x} \), \( -3x \) flips the direction, converting it to a large positive number. This means that the function heads towards positive infinity.

Understanding infinity in calculus allows us to determine whether certain limits are finite or boundless. Knowing this enables us to make meaningful interpretations of mathematical behaviors in real-world and abstract scenarios.

Finite Limits

Finite limits in calculus occur when a function approaches a particular, constant value as the input nears a specified point. Unlike infinite limits, finite limits suggest a converging behavior towards a tangible number.

• For instance, the limit \( \lim_{x \to -\infty} e^{3x} = 0 \) illustrates a finite limit. As \( x \) moves further into negative territory, this exponential function approaches and converges to the value zero.
• However, not all limits yield finite results. As seen with \( \lim_{x \to -\infty} e^{-3x} \), which tends toward infinity, there is no finite convergence.

Finite limits provide a framework for understanding how functions behave at extremes or boundaries, which is fundamental in analyzing continuity, convergence, and the behavior within different calculus applications. Recognizing whether a limit is finite supports insights into function properties and helps to solve complex practical and theoretical problems.