Problem 72 - Thomas Calculus in SI Units

Question

L'Hópital's Rule does not help with the limits. Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow-\infty} \frac{2^{x}+4^{x}}{5^{x}-2^{x}}$$

Step-by-Step Solution

Answer

The limit is \( \infty \) as \( x \to -\infty \).

Step 1: Identify the Dominant Terms in the Numerator and Denominator

In the expression \( \frac{2^{x} + 4^{x}}{5^{x} - 2^{x}} \), as \( x \to -\infty \), the exponential terms with smaller bases decrease faster. Therefore, identify the largest bases: \(4^x\) in the numerator and \(5^x\) in the denominator.

Step 2: Simplify the Expression by Dominant Terms

Rewrite the limit using the dominant terms from each part. Thus, simplify \( \frac{2^{x} + 4^{x}}{5^{x} - 2^{x}} \) approximately to \( \frac{4^{x}}{5^{x}} \) as \( x \to -\infty \), because the other terms diminish more rapidly.

Step 3: Rewrite the Fractions in Base 10

Express \(4^x\) as \((2^2)^x = 2^{2x}\) and \(5^x\) directly in the base form. The simplified expression is \( \frac{(2^2)^x}{5^x} = \left(\frac{2^2}{5}\right)^x = \left(\frac{4}{5}\right)^x\).

Step 4: Evaluate the Limit of the Simplified Expression

Now, consider the expression \( \left(\frac{4}{5}\right)^x\) as \( x \to -\infty \). The base \( \frac{4}{5} < 1 \) and when taken to a negative large power, \( \left(\frac{4}{5}\right)^x \to \infty\).

Step 5: Conclude the Result

Given that \( \frac{4}{5} \) raised to a negative power results in the reciprocal being large, the limit \( \left(\frac{4}{5}\right)^x \to \infty\) as \( x \to -\infty\). This means the original expression grows without bounds.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions involving numbers called bases raised to a power or exponent. These functions appear frequently in real-world applications like population growth, radioactive decay, and interest calculations. Key features include:

• Base: The number that's repeatedly multiplied. Common bases are numbers such as 2, 4, and 5, as seen in our exercise.
• Exponent: The power to which the base is raised. For instance, in the term \(2^x\), \(x\) is the exponent.
• Behavior at Infinity: When an exponential function has a base greater than 1, it grows rapidly as \(x\) increases. Conversely, if the base is less than 1, it diminishes towards zero as \(x\) advances to high values.

Understanding exponential functions is crucial for interpreting which terms might dominate in expressions as variables approach infinity or negative infinity.

Dominant Terms

In calculus, dominant terms in a mathematical expression are the ones that grow fastest or decrease slowest as a variable tends towards infinity. Identifying these terms simplifies evaluating limits, especially when faced with expressions involving exponents. Steps to identify dominant terms:

• Look for the largest base in exponential functions. Here, a base denotes the underlying number in a term like \(b^x\). The term with the largest base becomes dominant as it decides the behavior of the whole expression for very large or small values of \(x\).
• Ignore smaller bases as their effects diminish quickly. For example, in the expression \(2^x + 4^x\), the term \(4^x\) will dominate as \(x\) becomes very negative because \(2^x\) reduces more rapidly.
• Simplify the expression by retaining dominant terms only, which makes calculations and interpretations easier.

Understanding dominant terms is crucial for accurately evaluating the limits and tracking how different terms affect each other as they tend towards extremities.

Limits at Infinity

In calculus, studying limits at infinity involves analyzing what happens to a function as the variable approaches positive or negative infinity. This approach helps us understand the long-term behavior of mathematical expressions.Key concepts include:

• Approaching Infinity: When you take the limit as \(x \to \infty\) or \(x \to -\infty\), you're exploring what values the function seems to approach.
• Growth vs. Decay: For a term with a base greater than 1, like \(4^x\), it diminishes towards zero as \(x \to -\infty\), making these terms often negligible in such cases.
• Convergence: In the example \((\frac{4}{5})^x\), where the base is less than 1, the function grows rapidly towards infinity as \(x \to -\infty\), indicating the expression's rate of convergence.

Mastering limits at infinity helps not only in calculus but also gives insights into complex scenarios, like the behavior of series, improving our understanding of mathematical models.