\lim_{x \to +\infty}{(\frac{x+1}{x+2})^x}

We are tasked with finding:

\[
\lim_{x \to +\infty} \left( \frac{x+1}{x+2} \right)^x
\]

### Step 1: Simplify the base expression

We can rewrite the base expression:

\[
\frac{x+1}{x+2} = 1 - \frac{1}{x+2}
\]

Thus, the limit becomes:

\[
\lim_{x \to +\infty} \left( 1 - \frac{1}{x+2} \right)^x
\]

### Step 2: Apply logarithms

To deal with the exponent \(x\), let's take the natural logarithm of the expression. Let:

\[
L = \lim_{x \to +\infty} \left( 1 - \frac{1}{x+2} \right)^x
\]

Now take the natural logarithm of both sides:

\[
\ln(L) = \lim_{x \to +\infty} x \ln\left( 1 - \frac{1}{x+2} \right)
\]

### Step 3: Use the approximation for \(\ln(1 - y)\)

For small \(y\), we know that \(\ln(1 - y) \approx -y\). Here, \(y = \frac{1}{x+2}\), which becomes very small as \(x \to +\infty\). Therefore, we approximate:

\[
\ln\left( 1 - \frac{1}{x+2} \right) \approx -\frac{1}{x+2}
\]

Thus, the logarithmic limit becomes:

\[
\ln(L) = \lim_{x \to +\infty} x \left( -\frac{1}{x+2} \right)
\]

### Step 4: Simplify the limit

As \(x \to +\infty\), \(x+2 \approx x\), so we have:

\[
\ln(L) \approx \lim_{x \to +\infty} x \left( -\frac{1}{x} \right) = -1
\]

### Step 5: Exponentiate to find \(L\)

Now, exponentiate both sides to find \(L\):

\[
L = e^{-1} = \frac{1}{e}
\]

### Conclusion:

The limit is:

\[
\lim_{x \to +\infty} \left( \frac{x+1}{x+2} \right)^x = \frac{1}{e}
\]

Would you like more details or clarifications on any step?

Here are 5 related questions:
1. How can logarithms simplify limits involving exponents?
2. Why does the approximation \(\ln(1 - y) \approx -y\) hold for small \(y\)?
3. What are other methods for evaluating limits of the form \(\lim_{x \to +\infty} f(x)^x\)?
4. How does the behavior of functions at infinity affect their limits?
5. How can the binomial expansion help in evaluating limits?

**Tip:** When dealing with limits that involve expressions raised to infinity, logarithms can often simplify the calculations considerably.

\lim_{x \to +\infty}{\left(\frac{x+1}{x+2}\right)^x}

Explore the step-by-step solution to the limit of (x+1)/(x+2) raised to the power of x as x approaches infinity. This problem uses key calculus concepts such as logarithmic approximation and exponential functions, providing a clear breakdown of each step. Suitable for advanced high school and university students.

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