The value of $lim_{x\rightarrow\infty}(\frac{x+6}{x+1})^{x+4}$ is:

Q: The value of $lim_{x\rightarrow\infty}(\frac{x+6}{x+1})^{x+4}$ is:

To determine the value of $\lim_{x\rightarrow\infty}\left(\frac{x+6}{x+1}\right)^{x+4}$, we need to analyze the expression and simplify it as $x$ approaches infinity. 1. **Simplifying the Base:** \[ \frac{x+6}{x+1} = \frac{x(1 + \frac{6}{x})}{x(1 + \frac{1}{x})} = \frac{1 + \frac{6}{x}}{1 + \frac{1}{x}} \] As $x \to \infty$, $\frac{6}{x} \to 0$ and $\frac{1}{x} \to 0$, so the expression simplifies to: \[ \frac{1 + 0}{1 + 0} = 1 \] 2. **Applying the Exponent:** The expression becomes: \[ \left(\frac{x+6}{x+1}\right)^{x+4} = \left(1 + \frac{5}{x+1}\right)^{x+4} \] We recognize this as a form that resembles the limit definition of $e$, where: \[ \left(1 + \frac{1}{n}\right)^n \to e \quad \text{as} \quad n \to \infty \] 3. **Transforming the Expression:** To match the form $\left(1 + \frac{1}{n}\right)^n$, we set: \[ n = \frac{x+1}{5} \quad \Rightarrow \quad \frac{5}{x+1} = \frac{1}{n} \] Thus, the expression becomes: \[ \left(1 + \frac{1}{n}\right)^{5n \cdot \frac{x+4}{x+1}} \] 4. **Evaluating the Limit:** As $x \to \infty$, $\frac{x+4}{x+1} \to 1$, so the expression simplifies to: \[ \left(1 + \frac{1}{n}\right)^{5n} \] Which approaches $e^5$ as $n \to \infty$. Therefore, the value of the limit is indeed $e^5$, which corresponds to Option 4. **Explanation of Incorrect Options:** - **Option 1: $e^2$**, **Option 2: $e^3$**, and **Option 3: $e^4$** are incorrect because the transformation and simplification of the expression clearly lead to the exponent being 5, not 2, 3, or 4. The structure of the limit and the behavior of the expression as $x$ approaches infinity specifically result in the exponent being 5, which is why these options do not match the derived result. In conclusion, the correct answer is Option 4: $e^5$, as demonstrated by the detailed analysis and simplification of the given limit expression.

Question

Accepted Answer

To determine the value of $\lim_{x\rightarrow\infty}\left(\frac{x+6}{x+1}\right)^{x+4}$, we need to analyze the expression and simplify it as $x$ approaches infinity.

1. **Simplifying the Base:**
   $$
   \frac{x+6}{x+1} = \frac{x(1 + \frac{6}{x})}{x(1 + \frac{1}{x})} = \frac{1 + \frac{6}{x}}{1 + \frac{1}{x}}
   $$
   As $x \to \infty$, $\frac{6}{x} \to 0$ and $\frac{1}{x} \to 0$, so the expression simplifies to:
   $$
   \frac{1 + 0}{1 + 0} = 1
   $$

2. **Applying the Exponent:**
   The expression becomes:
   $$
   \left(\frac{x+6}{x+1}\right)^{x+4} = \left(1 + \frac{5}{x+1}\right)^{x+4}
   $$
   We recognize this as a form that resembles the limit definition of $e$, where:
   $$
   \left(1 + \frac{1}{n}\right)^n \to e \quad \text{as} \quad n \to \infty
   $$

3. **Transforming the Expression:**
   To match the form $\left(1 + \frac{1}{n}\right)^n$, we set:
   $$
   n = \frac{x+1}{5} \quad \Rightarrow \quad \frac{5}{x+1} = \frac{1}{n}
   $$
   Thus, the expression becomes:
   $$
   \left(1 + \frac{1}{n}\right)^{5n \cdot \frac{x+4}{x+1}}
   $$

4. **Evaluating the Limit:**
   As $x \to \infty$, $\frac{x+4}{x+1} \to 1$, so the expression simplifies to:
   $$
   \left(1 + \frac{1}{n}\right)^{5n}
   $$
   Which approaches $e^5$ as $n \to \infty$.

Therefore, the value of the limit is indeed $e^5$, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1: $e^2$**, **Option 2: $e^3$**, and **Option 3: $e^4$** are incorrect because the transformation and simplification of the expression clearly lead to the exponent being 5, not 2, 3, or 4. The structure of the limit and the behavior of the expression as $x$ approaches infinity specifically result in the exponent being 5, which is why these options do not match the derived result.

In conclusion, the correct answer is Option 4: $e^5$, as demonstrated by the detailed analysis and simplification of the given limit expression.