Guess the value of the limit by evaluating the function at values close to where familiar.) lim _(xarrow 0)(1+x)^1/x square

To evaluate the limit \(\lim_{x \to 0} (1+x)^{1/x}\), we can use the fact that this expression is a well-known limit in calculus. The limit of \((1+x)^{1/x}\) as approaches 0 is the definition of the number , the base of the natural logarithm.To understand why this is true, consider the following steps:1. **Rewrite the Expression Using Exponential Functions:** 2. **Analyze the Argument of the Exponential Function:** We need to find the limit of \(\frac{\ln(1+x)}{x}\) as approaches 0.3. **Use the Taylor Series Expansion:** For small values of , \(\ln(1+x)\) can be approximated by its Taylor series: Therefore, 4. **Take the Limit:** As approaches 0, the terms involving in the expansion go to 0: 5. **Combine the Results:** Since \(\frac{\ln(1+x)}{x}\) approaches 1 as approaches 0, we have: Therefore, the value of the limit is: