Problem 70 - Calculus

Question

In Problems \(64-71\), find a value of the constant \(k\) such that the limit exists. $$\lim _{x \rightarrow \infty} \frac{3^{k x}+6}{3^{2 x}+4}$$

Step-by-Step Solution

Answer

The value of \(k\) is 2.

Step 1: Express Terms with Common Bases

The given expression is \( \lim_{x \to \infty} \frac{3^{kx} + 6}{3^{2x} + 4} \). To simplify the expression, note that 6 and 4 are negligible compared to 3 raised to any power as \( x \) approaches infinity. Therefore, the dominant terms are \(3^{kx}\) in the numerator and \(3^{2x}\) in the denominator.

Step 2: Simplify Dominant Terms

Rewrite the fraction focusing on the exponents of 3. The expression becomes: \( \lim_{x \to \infty} \frac{3^{kx}}{3^{2x}} = \lim_{x \to \infty} 3^{kx - 2x} = \lim_{x \to \infty} 3^{(k-2)x} \). We seek a value of \( k \) where the limit exists.

Step 3: Analyze the Limit Based on the Value of k

The term \(3^{(k-2)x}\) approaches a finite number if \(k-2 = 0\) because otherwise, it increases to infinity or diminishes to zero. This implies that \(k-2 = 0\) or \(k = 2\).

Step 4: Verify the Value of k

If \( k = 2 \), then \(3^{(k-2)x} = 3^0 = 1\). This confirms that the limit exists and is equal to \(1\) when \(k = 2\).

Key Concepts

Dominant Terms

When solving limit problems, especially those approaching infinity, identifying the dominant terms in the expression can significantly simplify the problem. Dominant terms are the parts of the equation that have the greatest influence as the variable approaches a particular value, often infinity. In the context of our exercise, we're dealing with exponential terms involving powers of 3, such as \(3^{kx}\) and \(3^{2x}\).

As \(x\) tends towards infinity, lower-order terms like \(6\) in the numerator and \(4\) in the denominator become irrelevant because their impact is minuscule compared to exponential growth. Thus, in the expression \( \frac{3^{kx} + 6}{3^{2x} + 4} \), the dominant terms are \(3^{kx}\) and \(3^{2x}\).

Identifying and focusing on these dominant terms allows us to simplify the limit to \( \frac{3^{kx}}{3^{2x}} \), making it easier to analyze and ultimately leading us to find a value for \(k\) that makes the limit finite.

Exponential Functions

Exponential functions are significant in calculus for their unique properties. They involve expressions where a constant base is raised to a variable exponent. For instance, functions like \(3^x\) are exponential, and they increase (or decrease) exceptionally fast compared to polynomial or linear functions.

One of the critical properties of exponential functions is their growth behavior. As mentioned in our context, exponential terms \(3^{kx}\) and \(3^{2x}\) grow dramatically larger as \(x\) increases. This growth can overshadow other terms in an equation, highlighting the importance of dominant terms analyses in limits.

• Exponential decay occurs when the base is a fraction or when the exponent is negative.
• The base being greater than 1 leads to exponential growth, such as in our expression.

Understanding these properties helps determine how exponential functions influence the value of the limit as \(x\) approaches infinity, which is crucial when exploring limits of these functions.

Infinity Limits

Infinity limits arise when a variable within a function approaches infinity, and we're interested in what happens to the function's values. These limits are pivotal in understanding the behavior of functions at their extremes.

The limit \( \lim_{x \to \infty} 3^{(k-2)x} \) addressed in our exercise is an excellent example of exploring infinity limits. Such an expression reveals that depending on the value of \(k\), the function can:

• Approach zero (if \(k < 2\)) because the negative exponent implies exponential decay.
• Approach infinity (if \(k > 2\)) due to exponential growth.
• Settle at a finite value (if \(k = 2\)), resulting in \(3^0 = 1\).

From this analysis, the insights gleaned can predict and determine the function's long-term behavior as \(x\) continues to grow. This understanding exemplifies the power and necessity of evaluating limits as they approach infinity in calculus.