Let f(x) = (1 + x)1⁄x.(a)Estimate the value of the limit lim x→0 (1 + x)1⁄x to five decimal places.

1. Break Down the Problem

We need to estimate the limit: $\lim_{x \to 0} (1 + x)^{1/x}$

2. Relevant Concepts

To evaluate the limit as $x$ approaches 0, we can use the property that: $\lim_{x \to 0} (1 + x)^{1/x} = e$ This is derived from the definition of the number $e$.

3. Analysis and Detail

Using the natural logarithm:

1. We can take the natural logarithm of the function: $\ln y = \ln((1 + x)^{1/x}) = \frac{1}{x} \ln(1 + x)$

2. We can evaluate the limit of $\ln y$: $\lim_{x \to 0} \ln y = \lim_{x \to 0} \frac{\ln(1 + x)}{x}$

3. Applying L'Hôpital's Rule since the limit is of the form $\frac{0}{0}$: $= \lim_{x \to 0} \frac{\frac{d}{dx} \ln(1 + x)}{\frac{d}{dx} x}$ Substituting derivatives: $= \lim_{x \to 0} \frac{\frac{1}{1 + x}}{1} = \lim_{x \to 0} \frac{1}{1 + x} = 1$

4. Therefore: $\lim_{x \to 0} \ln y = 1 \implies y = e^1 = e$

4. Verify and Summarize

Thus, the limit evaluated is: $\lim_{x \to 0} (1 + x)^{1/x} = e \approx 2.71828$

Final Answer

The estimated value of the limit $\lim_{x \to 0} (1 + x)^{1/x}$ to five decimal places is: $\boxed{2.71828}$