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Chapter 4: Problem 48

In Problems \(48-50,\) evaluate the limit using the fact that $$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e$$ $$\lim _{x \rightarrow 0^{+}}(1+x)^{1 / x}$$

Short Answer

Expert verified
The limit is \( e \).

Step by step solution

01

Understand the Problem

We are asked to evaluate the limit \( \lim _{x \rightarrow 0^{+}}(1+x)^{1 / x} \). It is important to recognize that this expression takes a form similar to the known limit \( \lim _{n \to \infty}\left(1+\frac{1}{n}\right)^{n}=e \).
02

Substitute Variable

To make the expression fit the form \(\left(1+\frac{1}{n}\right)^{n}=e\), let \(y = \frac{1}{x}\). As \(x\) approaches \(0^{+}\), \(y\) approaches \(\infty\).
03

Rewrite the Expression

Rewrite the original limit using the substitution: \( \lim_{x \to 0^{+}} (1 + x)^{1/x} \) becomes \( \lim_{y \to \infty} (1 + 1/y)^{y} \).
04

Apply Known Limit Result

Recognize the expression \( \lim_{y \to \infty} (1 + 1/y)^{y} \) as directly matching the known form \( \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n} = e \). Thus, the limit evaluates to \( e \).
05

Conclusion

Since the rewritten expression directly matches the well-established limit, we conclude that\[ \lim _{x \rightarrow 0^{+}}(1+x)^{1 / x} = e. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Evaluation
Evaluating limits is a fundamental process in calculus. It involves finding the value that a function approaches as the variable tends toward a particular point or infinity. In the problem given, the function to evaluate is \( \lim _{x \rightarrow 0^{+}}(1+x)^{1 / x} \). Understanding limit evaluation begins with recognizing how the expression behaves as \( x \) approaches the specified condition.
  • First, identify the form of the limit, as it may resemble standard forms you know.
  • Next, apply well-known results or rules to simplify the expression.
  • Lastly, ensure the transformation is valid under the limit鈥檚 conditions.
This particular limit is similar to others in calculus, such as approaching values at infinity or adjusting tiny changes near zero. Effective limit evaluation often relies on substitutions or recognizing special forms.
Variable Substitution
Variable substitution is a helpful method to simplify limits by transforming expressions into familiar forms. In this problem, we begin by setting \( y = \frac{1}{x} \). This substitution dramatically simplifies our work. Here鈥檚 how it helps:When you substitute variables, keep in mind the direction and the value where the original variable tends.
  • As \( x \to 0^{+} \), the substitution means \( y \to \infty \).
  • This change alters the original limit \( \lim_{x \to 0^{+}} (1 + x)^{1/x} \) to \( \lim_{y \to \infty} (1 + 1/y)^{y} \).
  • This substitution converts the tricky expression into a familiar one that matches a well-known pattern.
Variable substitution isn鈥檛 just a strategy in calculus; it's like translating the problem into a language or form that is easier to analyze and solve.
Exponential Limit
The concept of exponential limits is embedded deeply within calculus, particularly when exploring growth patterns and continuous compounding. The expression \( \lim _{n \rightarrow \infty}(1+\frac{1}{n})^{n} = e \) is a cornerstone in this area.The given problem pivots around this known exponential limit.
  • By using the substitution \( y = \frac{1}{x} \), the expression becomes \( \lim_{y \to \infty} (1 + 1/y)^{y} \).
  • This is exactly the structure of the exponential limit that evaluates to \( e \).
  • The substitution illustrates how seemingly different limits relate to exponential growth concepts, leveraging existing knowledge for quick evaluation.
Exponential limits often describe processes where quantities grow steadily at rates proportional to their current size, hinting at notions like the number \( e \) which appears frequently in nature and finance.

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