Problem 50 - Calculus for Biology and Medicine

Question

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty}(1+x)^{1 / x} $$

Step-by-Step Solution

Answer

The limit of \((1+x)^{1/x}\) as \(x\) approaches infinity is 1.

Step 1: Understanding the Limit Expression

We need to find the limit of the expression \((1+x)^{1/x}\) as \(x\) approaches infinity. The base \((1+x)\) grows indefinitely while the exponent \(\frac{1}{x}\) tends towards zero.

Step 2: Recalling a Relevant Limit Result

Recall that for large values of \(x\), \((1+x)^{1/x}\) can be reinterpreted using the limit definition: \(\lim_{x \to \infty} (1+\frac{1}{x})^x = e\). This can be transformed into our expression.

Step 3: Transforming the Original Expression

Rewrite \((1+x)^{1/x}\) as \((1+\frac{1}{x})^x\), since dividing \(x\) inside \((1+x)\) is equivalent to the division of the base expression by \(x\).

Step 4: Applying Known Limit Results

Using the earlier mentioned result, as \(x\) approaches infinity, \((1+\frac{1}{x})^x\) approaches \(e\). Therefore, our limit \((1+x)^{1/x}\) should tend towards \(1\), not \(e\), as \(x\) goes to infinity.

Step 5: Final Conclusion

After analysis, the correct interpretation of \((1+x)^{1/x}\) as \(x\rightarrow \infty\) reflects the constants more correctly as tending to \(1\) than supporting a limit evaluation of \(e\).

Key Concepts

Understanding l'Hôpital's Rule

l'Hôpital's Rule is a handy and powerful tool for evaluating certain types of limits. Typically, it is used when direct substitution in a limit leads to an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). In such cases, l'Hôpital's Rule states that:
\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \]

• This holds true provided the limit on the right-hand side exists or is infinite.
• "a" can be any real number, infinity, or negative infinity.

When applying l'Hôpital's Rule, make sure to check that both the function's numerator and denominator result in an indeterminate form. In the context of our original problem, direct application of l'Hôpital's Rule wasn't necessary as the expression "rewriting" handled the indeterminate nature, leading to a direct result.

Navigating Infinity Limits

When dealing with limits approaching infinity, the behavior of the function can sometimes be unpredictable or counter-intuitive. Instead, it's crucial to understand how different expressions "behave" as variables approach infinity. Oftentimes:

• Polynomial expressions and terms typically grow larger faster than exponential bases with diminishing exponents.
• Expressions approaching forms like \( \left( 1 + \frac{1}{x} \right)^x \) can lead us to constants like "e" when transformations are properly done.

Understanding how expressions can change or reduce helps in evaluating these prestigious infinity limits wisely. As we saw in the original step-by-step process, converting the original base and exponent allowed for an easy application of known limit results.

Decoding Exponential Expressions

Exponential expressions, such as \((1+x)^{1/x}\), reveal fascinating behaviors when their terms both grow and shrink. The exponential base here, \(1+x\), heads off to infinity while the exponent \(1/x\) diminishes toward zero. Such expressions are typical in limit problems where understanding base-exponent interactions is crucial.

In some cases:

• Even though the base tends to infinity, a diminishing exponent can converge the entire expression to a particular value.
• This is vividly illustrated with the transformation to \((1+\frac{1}{x})^x\), leading to a result anchored at "e", recognizing the essential behavior often seen with such structures.

Grasping these interactions and transformations is central to unlocking the sometimes hidden secrets within exponential expressions encountered in calculus limit problems.