To solve for [tex]\( y \)[/tex] in the function [tex]\( y = \left(1 + \frac{1}{x}\right)^x \)[/tex], we need to evaluate this expression at different values of [tex]\( x \)[/tex]. Rounding the results to the nearest thousandth, we create the following table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 2 \\
\hline
10 & 2.594 \\
\hline
100 & 2.705 \\
\hline
10,000 & 2.718 \\
\hline
100,000 & 2.718 \\
\hline
1,000,000 & 2.718 \\
\hline
\end{array}
\][/tex]
This table shows us the values of [tex]\( y \)[/tex] for different values of [tex]\( x \)[/tex], rounded to the nearest thousandth:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex]
- When [tex]\( x = 10 \)[/tex], [tex]\( y \approx 2.594 \)[/tex]
- When [tex]\( x = 100 \)[/tex], [tex]\( y \approx 2.705 \)[/tex]
- When [tex]\( x = 10,000 \)[/tex], [tex]\( y \approx 2.718 \)[/tex]
- When [tex]\( x = 100,000 \)[/tex], [tex]\( y \approx 2.718 \)[/tex]
- When [tex]\( x = 1,000,000 \)[/tex], [tex]\( y \approx 2.718 \)[/tex]
Thus, the rounded values are:
[tex]\[
a \approx 2.594 \\
b \approx 2.705 \\
c \approx 2.718 \\
d \approx 2.718 \\
e \approx 2.718
\][/tex]
From these calculations, the values of [tex]\( y \)[/tex] become very close to [tex]\( e \)[/tex] (approximately 2.718) as [tex]\( x \)[/tex] increases, illustrating the concept of the limit of the expression as [tex]\( x \)[/tex] approaches infinity.
