Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each limit. (a) \(\lim _{x \rightarrow 0^{+}}\left(1^{x}+2^{x}\right)^{1 / x}\) (b) \(\lim _{x \rightarrow 0^{-}}\left(1^{x}+2^{x}\right)^{1 / x}\) (c) \(\lim _{x \rightarrow \infty}\left(1^{x}+2^{x}\right)^{1 / x}\) (d) \(\lim _{x \rightarrow-\infty}\left(1^{x}+2^{x}\right)^{1 / x}\)

Step-by-Step Solution

Answer

(a) \( \infty \), (b) \( 0 \), (c) 2, (d) 1.

Step 1: Analyze part (a)

We need to find \( \lim _{x \rightarrow 0^{+}} \left(1^{x}+2^{x}\right)^{1 / x} \). As \( x \) approaches 0 from the right, the term \( 2^x \rightarrow 1 \) since any number to the power of 0 is 1. Thus, \( 1^x+2^x \rightarrow 1 + 1 = 2 \). The expression becomes \( \left(2\right)^{1/x} \). Since \( \frac{1}{x} \) approaches infinity as \( x \) approaches 0 from the positive side, the expression goes to \( \infty \).

Step 2: Analyze part (b)

We now find \( \lim _{x \rightarrow 0^{-}} \left(1^{x}+2^{x}\right)^{1 / x} \). As \( x \) approaches 0 from the negative side, \( 2^x \rightarrow 1 \), similar to the positive case, and \( 1^x+2^x \rightarrow 1 + 1 = 2 \) again. The expression becomes \( \left(2\right)^{1/x} \). However, since \( \frac{1}{x} \) approaches negative infinity when \( x \rightarrow 0^{-} \), the result is \( 0 \).

Step 3: Analyze part (c)

Now we find \( \lim _{x \rightarrow \infty} \left(1^{x}+2^{x}\right)^{1 / x} \). As \( x \) approaches infinity, \( 1^x = 1 \) remains constant, and \( 2^x \rightarrow \infty \). Thus, \( 1^x+2^x \approx 2^x \). The expression simplifies as \( \left(2^x\right)^{1/x} = 2^{x/x} = 2 \).

Step 4: Analyze part (d)

Lastly, find \( \lim _{x \rightarrow -\infty} \left(1^{x}+2^{x}\right)^{1 / x} \). As \( x \rightarrow -\infty \), both \( 1^x \rightarrow 1 \) and \( 2^x \rightarrow 0 \). Hence, \( 1^x+2^x \rightarrow 1 \). The expression becomes \( 1^{1/x} \), which approaches \( 1 \) since any number raised to the power that vanishes approaches 1.

Key Concepts

One-Sided Limits

When we talk about one-sided limits, we're essentially looking at what happens to a function as it approaches a certain point, but only from one direction. Let's consider two scenarios: approaching from the left (negative side) and from the right (positive side). For example, in the exercise, part (a) calculates the limit as x approaches 0 from the positive side, noted as \( x \to 0^{+} \), while part (b) approaches 0 from the negative side, or \( x \to 0^{-} \).

• From the positive side \( x \to 0^{+} \): Here, \( x \) gets closer to 0 but is always greater than 0.
• From the negative side \( x \to 0^{-} \): \( x \) approaches 0 but is always less than 0.

In solving limits, especially when dealing with functions like \( (1^x + 2^x)^{1/x} \), the direction from which \( x \) approaches the value can significantly affect the limit result. One-sided limits help in understanding these subtle differences in behavior.

Infinity Limits

Infinity limits describe how a function behaves as the input \( x \) moves towards positive or negative infinity. In the context of the textbook problem, consider parts (c) and (d). - For part (c), \( x \to \infty \): As \( x \) becomes very large, terms like \( 2^x \) grow rapidly, overwhelming constants like \( 1^x = 1 \). This leads us to the limit where \( (1^x + 2^x)^{1/x} \) simplifies dominantly by \( 2^x \) giving us \( 2 \) as the limit.- For part (d), \( x \to -\infty \): Here, \( x \) is heading towards a very negative value, causing terms like \( 2^x \) to approach \( 0 \), while \( 1^x = 1 \) remains. The function behaves such that \( (1^x + 2^x)^{1/x} \) approximates to \( 1 \).Infinity limits examine these extreme behaviors, facilitating our understanding of how a function grows or declines towards infinity.

Exponential Functions

Exponential functions are fundamental in these exercises. An exponential function is of the form \( a^x \), where \( a \) is a constant base, and \( x \) is the exponent. In problems such as those in the exercise, different behaviors are exhibited as \( x \) changes:

• For positive \( x \), \( a^x \) increases rapidly if \( a > 1 \), like \( 2^x \).
• For negative \( x \), \( a^x \) becomes a fraction and approaches zero.
• At \( x = 0 \), \( a^x = 1 \).

Understanding the nature of exponential growth and decay is critical for correctly applying limits, as shown in the problem solutions, where exponential terms influence the limit's behavior in varied ways.

Step-by-Step Solutions

Step-by-step solutions guide you through the process of solving complex problems systematically. Let's see how they help clarify the solution to limit problems like our exercise:

• Identify the Limit Problem: Start by identifying the type of limit problem: one-sided, infinity, or involving an exponential function.
• Break Down the Problem: Analyze each part separately, considering specific conditions such as \( x \to 0^{+} \) or \( x \to -\infty \).
• Simplify Expressions: Simplify by focusing on major contributing factors in functions like \( (1^x + 2^x)^{1/x} \).
• Evaluate the Limit: Finally, calculate the limit using known properties and simplifications you've identified.

These detailed steps empower you to tackle similar problems by applying the techniques uniformly, ensuring a deeper understanding through careful, deliberate practice.