Problem 34 - Advanced Engineering Mathematics

Question

As \(x \rightarrow-\infty, e^{6 x} \rightarrow 0\) and \(y \rightarrow 2 x+3 .\) Now write \(\left(1+c e^{6 x}\right) /\left(1-\alpha e^{6 x}\right)\) as \(\left(e^{-6 x}+c\right) /\left(e^{-6 x}-c\right) .\) Then, as \(x \rightarrow \infty, e^{-6 x} \rightarrow 0\) and \(y \rightarrow 2 x-3\).

Step-by-Step Solution

Answer

As \(x \to -\infty\), \(y \to 2x+3\). As \(x \to \infty\), \(y \to 2x-3\).

Step 1: Understanding the Behavior at Negative Infinity

As we consider the behavior of the exponential function, notice that as \(x \rightarrow -\infty\), the expression \(e^{6x}\) approaches 0. In this scenario, the term \(y\) simplifies to \(2x + 3\). We need to find the corresponding behavior when expressed with \(e^{-6x}\).

Step 2: Manipulating the Expression

We are given the expression \(\frac{1+ce^{6x}}{1-\alpha e^{6x}}\). To transform it, let's multiply both numerator and denominator by \(e^{-6x}\). This multiplication is essentially dividing each term by \(e^{6x}\):

Step 3: Multiplying by \(e^{-6x}\)

Multiply the terms: \(\left(1 + ce^{6x}\right) e^{-6x} = e^{-6x} + c\) and \(\left(1 - \alpha e^{6x}\right)e^{-6x} = e^{-6x} - \alpha\). Thus, we rewrite the expression as \(\frac{e^{-6x} + c}{e^{-6x} - \alpha}\).

Step 4: Understanding the Behavior at Positive Infinity

As \(x \rightarrow \infty\), note that \(e^{-6x}\) approaches 0. In this altered expression, \(y\) simplifies to \(2x - 3\). This is because the exponential term vanishes, reflecting a change in the offset of the linear term.

Key Concepts

Limits and Behavior at Infinity

In mathematics, a **limit** describes the behavior of a function as the input approaches a particular value. When we examine functions at infinity, we're assessing how a function behaves as the variable grows infinitely large or small. Consider the exponential function

• As \( x \to -\infty \), the expression \( e^{6x} \to 0 \). This means that the value of the function diminishes towards zero, effectively disappearing as we move left on the number line.
• Conversely, as \( x \to \infty \), the expression \( e^{-6x} \to 0 \). Here, the exponential decay results in the function diminishing from zero as we move rightward on the number line.

In practice, these limits help determine the resulting behavior of the function or expression as the input becomes exceedingly large or small. This is crucial when simplifying expressions to find approximate values or behaviors without the overshadowing influence of remaining components.

Expression Manipulation

**Expression manipulation** involves creatively altering mathematical expressions to reveal or simplify their structure. In our scenario, we start with the expression \( \frac{1+c e^{6x}}{1-\alpha e^{6x}} \).
Notice that multiplying the entire expression by \( e^{-6x} \) is a strategic move. It effectively changes the basis of the expression:

• For the numerator: \( (1+ce^{6x})e^{-6x} = e^{-6x} + c \).
• For the denominator: \( (1-\alpha e^{6x})e^{-6x} = e^{-6x} - \alpha \).

Multiplying by \( e^{-6x} \) aligns the exponential powers, simplifying the transformation. This approach stems from wanting to analyze the behavior at positive infinity, making it more intuitive. Manipulation aids in restructuring problems to highlight aspects like limits or simplification, essential strategies for algebraic problem-solving.

Exponential Decay

**Exponential decay** is a process where a quantity decreases over time, proportionally to its current value. In our exercise, this concept can be observed in

• As \( x \to \infty \), \( e^{-6x} \to 0 \) depicts decay since the term diminishes quickly.

This behavior is inherent in many real-world scenarios, such as radioactive decay, depreciation, and cooling laws.
The characteristic feature of exponential decay is its rapid drop-off—once a base element decreases, it's multiplied by a constant factor each time period, further reducing its overall presence. This type of decay is indicative of inverse relationships, where large movements in \( x \) result in small resulting values for the associated function. When expressed in a modified function, this allows for simpler calculation by negating terms that become negligible beyond a certain boundary.
Understanding exponential decay is crucial in many fields like physics, finance, and population studies, and helps predict how quantities change over time.