Let's solve the given problem step by step.

The problem asks us to compute the limit:

$\lim_{x \to \infty} \left( \frac{x+5}{x+1} \right)^x$

and express the result in the form $e^a$, where $a = \ldots$.

Step 1: Rewrite the base

The base of the expression is: $\frac{x+5}{x+1}$ Divide numerator and denominator by $x$ (the highest power of $x$): $\frac{x+5}{x+1} = \frac{1 + \frac{5}{x}}{1 + \frac{1}{x}}$ As $x \to \infty$, the fractions $\frac{5}{x}$ and $\frac{1}{x}$ approach $0$. So: $\frac{x+5}{x+1} \to 1$

Step 2: Analyze the limit in exponential form

The original expression becomes: $\lim_{x \to \infty} \left( \frac{x+5}{x+1} \right)^x$ Let: $y = \left( \frac{x+5}{x+1} \right)^x$ Take the natural logarithm on both sides: $\ln y = x \ln \left( \frac{x+5}{x+1} \right)$

Step 3: Expand the logarithm

Using the approximation $\ln(1 + u) \approx u$ for small $u$: $\frac{x+5}{x+1} = 1 + \frac{4}{x+1}$ So: $\ln \left( \frac{x+5}{x+1} \right) \approx \frac{4}{x+1}$

Substitute back: $\ln y \approx x \cdot \frac{4}{x+1}$ Simplify the expression: $\ln y \approx \frac{4x}{x+1}$

Step 4: Compute the limit

As $x \to \infty$, $\frac{4x}{x+1} \to 4$. Thus: $\ln y \to 4$

Exponentiate to get $y$: $y \to e^4$

Final Answer:

$\boxed{e^4, \; a = 4}$

Would you like additional clarification or have further questions?
Here are 5 related questions to deepen your understanding:

1. How does the approximation $\ln(1 + u) \approx u$ work for small $u$?
2. Why do we take the natural logarithm when solving exponential limits?
3. How does dividing by $x$ simplify rational functions in limits?
4. Could the result change if the numerator or denominator had different constants?
5. Can you generalize the method to $\lim_{x \to \infty} \left( \frac{x+c}{x+d} \right)^x$?

Tip: When solving limits involving powers, try converting to logarithmic form early to simplify the calculation!