To determine which expression is closest in value to [tex]\( e \)[/tex], we need to evaluate each expression and compare them to the mathematical constant [tex]\( e \)[/tex], which is approximately 2.71828.
Let's evaluate each expression:
1. [tex]\(\left(1+\frac{1}{12}\right)^{12}\)[/tex]:
[tex]\[
\left(1+\frac{1}{12}\right)^{12} \approx 2.613035290224676
\][/tex]
2. [tex]\(\left(1+\frac{1}{11}\right)^{11}\)[/tex]:
[tex]\[
\left(1+\frac{1}{11}\right)^{11} \approx 2.6041990118975287
\][/tex]
3. [tex]\(\left(1+\frac{1}{14}\right)^{14}\)[/tex]:
[tex]\[
\left(1+\frac{1}{14}\right)^{14} \approx 2.6271515563008685
\][/tex]
4. [tex]\(\left(1+\frac{1}{13}\right)^{13}\)[/tex]:
[tex]\[
\left(1+\frac{1}{13}\right)^{13} \approx 2.6206008878857308
\][/tex]
Next, we find the absolute difference between each calculated value and [tex]\( e \approx 2.71828 \)[/tex]:
- For [tex]\( \left(1+\frac{1}{12}\right)^{12} \)[/tex]:
[tex]\[
|2.613035290224676 - 2.71828| \approx 0.105244709775324
\][/tex]
- For [tex]\( \left(1+\frac{1}{11}\right)^{11} \)[/tex]:
[tex]\[
|2.6041990118975287 - 2.71828| \approx 0.1140809881024713
\][/tex]
- For [tex]\( \left(1+\frac{1}{14}\right)^{14} \)[/tex]:
[tex]\[
|2.6271515563008685 - 2.71828| \approx 0.0911284436991315
\][/tex]
- For [tex]\( \left(1+\frac{1}{13}\right)^{13} \)[/tex]:
[tex]\[
|2.6206008878857308 - 2.71828| \approx 0.0976791121142692
\][/tex]
After evaluating the differences, we see that the smallest difference is for the expression [tex]\(\left(1+\frac{1}{14}\right)^{14}\)[/tex] with an absolute difference of approximately 0.0911284436991315.
Therefore, the value of the expression closest to [tex]\( e \)[/tex] is:
[tex]\[ \left(1+\frac{1}{14}\right)^{14} \][/tex]
which corresponds to choice C.
