Problem 58 - Calculus Early Transcendentals: Pearson New International Edition

Question

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 1^{+}} \frac{2}{1+2^{1 /(x-1)}} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Problem

We need to find the limit of the function \( f(x) = \frac{2}{1+2^{1/(x-1)}} \) as \( x \) approaches \( 1^+ \). This means evaluating what happens when \( x \) gets very close to 1 from the positive side.

Step 2: Simplify the Exponent

As \( x \rightarrow 1^+ \), the expression \( 1/(x-1) \) approaches infinity since the denominator approaches zero from the positive side.

Step 3: Analyze the Expression

Since \( 2^{1/(x-1)} \) will approach infinity as \( x \rightarrow 1^+ \), the expression \( 1 + 2^{1/(x-1)} \) also approaches infinity.

Step 4: Calculate the Limit

The fraction \( \frac{2}{1 + 2^{1/(x-1)}} \) becomes \( \frac{2}{\infty} \) as \( x \) approaches \( 1^+ \). Therefore, the limit of the function is \( 0 \).

Key Concepts

Software for Limits

Software programs can be invaluable tools when it comes to calculating limits in mathematics. These programs, like Mathematica and Wolfram Alpha, can handle complex calculations that would be cumbersome by hand. Nevertheless, it's important to remember that these programs are not always perfect and sometimes might give an incorrect result. It's essential to understand the underlying theory to verify the software's output.

When using software for limits, you can check their accuracy by testing them on known limits. If the software can accurately compute these, you can have more confidence in its results. However, if the result seems counterintuitive, consider analyzing the expression manually or using another method to confirm.

• Always double-check critical calculations.
• Use software as a tool for reinforcement, not as a crutch.

Ultimately, combining your understanding with software tools leads to a more robust mathematical ability.

Limits Approaching Infinity

When dealing with limits approaching infinity, you want to understand what happens to a function as it nears a specific point, often where the variable becomes very large or very small. In our case of \( x \rightarrow 1^+ \), this means approaching 1 from values slightly greater than 1.

The expression \( \frac{2}{1+2^{1/(x-1)}} \) provides an intriguing challenge. As \( x \) gets closer to 1, the denominator \( 1/(x-1) \) explodes towards infinity since you're dividing by a number getting closer and closer to zero. This changes the behavior of the exponential component \( 2^{1/(x-1)} \), since any positive number raised to a power approaching infinity becomes extremely large.

Thus, the fraction as a whole behaves like \( \frac{2}{\infty} \), simplifying to 0, since the denominator becomes infinitely large. Recognizing how components of an expression behave at these critical points is key to understanding limits.

Simplifying Expressions in Limits

Simplifying expressions within limits is a crucial step in understanding their behavior. Take, for example, the original function \( \frac{2}{1+2^{1/(x-1)}} \). The task is to simplify the expression before analyzing its limit.

Important steps in simplification involve assessing each component separately. When \( x \rightarrow 1^+ \), the term \( 1/(x-1) \) escalates to infinity, making the term \( 2^{1/(x-1)} \) swell astronomically large. Simplifying the expression helps us foresee the behavior as:\[\lim _{x \rightarrow 1^{+}} \frac{2}{1+2^{1/(x-1)}} = \frac{2}{\infty} = 0\]
Understanding the behavior of complex terms separately facilitates predicting the overall limit. As you practice more, recognizing which terms to focus on and how they influence the limit becomes more intuitive, guiding you to the correct conclusion.