Math Problem Statement

Find the limit using lhopitals and logarithms

Find the limit using L'Hôpital's Rule and logarithms for the expression \( \lim_{x \to \infty} \left( \frac{x}{x+1} \right)^x \).

Solution

To find the limit

limx(xx+1)x\lim_{x \to \infty} \left( \frac{x}{x+1} \right)^x

using L'Hôpital's Rule and logarithms, we can proceed with the following steps:

Step 1: Rewrite the Expression with Logarithms

Set y=(xx+1)xy = \left( \frac{x}{x+1} \right)^x. Taking the natural logarithm on both sides, we get: lny=xln(xx+1)\ln y = x \cdot \ln \left( \frac{x}{x+1} \right) Now, our goal is to find: limxlny=limxxln(xx+1)\lim_{x \to \infty} \ln y = \lim_{x \to \infty} x \cdot \ln \left( \frac{x}{x+1} \right)

Step 2: Simplify the Logarithmic Term

We can simplify ln(xx+1)\ln \left( \frac{x}{x+1} \right) as follows: ln(xx+1)=lnxln(x+1)\ln \left( \frac{x}{x+1} \right) = \ln x - \ln(x+1) Then, lny=x(lnxln(x+1))=xln(11x+1)\ln y = x \cdot \left( \ln x - \ln(x+1) \right) = x \cdot \ln \left(1 - \frac{1}{x+1}\right)

Step 3: Use a Taylor Expansion (for Small Terms)

For large xx, ln(11x+1)1x+1\ln \left(1 - \frac{1}{x+1}\right) \approx -\frac{1}{x+1}. Therefore, lnyx(1x+1)=xx+1\ln y \approx x \cdot \left( -\frac{1}{x+1} \right) = -\frac{x}{x+1}

Step 4: Take the Limit

Now we need to compute limxxx+1\lim_{x \to \infty} -\frac{x}{x+1} Dividing the numerator and denominator by xx, we get: limxxx+1=limx11+1x=1\lim_{x \to \infty} -\frac{x}{x+1} = \lim_{x \to \infty} -\frac{1}{1 + \frac{1}{x}} = -1

Step 5: Interpret the Result

Since lny1\ln y \to -1 as xx \to \infty, we have ye1=1ey \to e^{-1} = \frac{1}{e}.

Final Answer

limx(xx+1)x=1e\lim_{x \to \infty} \left( \frac{x}{x+1} \right)^x = \frac{1}{e}

Would you like more details on any of these steps, or do you have any questions?

Here are some related questions you might find useful:

  1. How do Taylor expansions help simplify limits involving logarithms?
  2. Why do we use L'Hôpital's Rule for indeterminate forms like ×0\infty \times 0?
  3. Can this problem be solved without using logarithms?
  4. What other methods exist for evaluating limits involving powers and logarithms?
  5. How would the limit change if we altered the power of xx in the expression?

Tip: In calculus, logarithmic transformations often simplify expressions by turning products and powers into sums and linear terms, making limits easier to evaluate.

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Math Problem Analysis

Mathematical Concepts

Limits
Logarithmic Transformations
Indeterminate Forms

Formulas

\( \lim_{x \to \infty} \left( \frac{x}{x+1} \right)^x \)
Taylor Expansion
L'Hôpital's Rule

Theorems

L'Hôpital's Rule

Suitable Grade Level

College Level Calculus