The image shows the following mathematical limit:

$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$

This is a well-known fundamental limit in calculus and forms the basis for the definition of the mathematical constant $e$, which is approximately $2.718$.

Explanation:

1. The expression $(1 + x)^{\frac{1}{x}}$ becomes problematic when directly substituting $x = 0$, since it results in an indeterminate form $1^ \infty$.
2. To evaluate the limit, we take the natural logarithm: $\ln L = \lim_{x \to 0} \frac{\ln(1 + x)}{x}$ where $L$ is the original limit expression.
3. Using the Taylor series expansion for $\ln(1 + x)$ around $x = 0$, we get: $\ln(1 + x) \approx x - \frac{x^2}{2} + \mathcal{O}(x^3)$
4. Substituting into the limit: $\ln L = \lim_{x \to 0} \frac{x - \frac{x^2}{2} + \dots}{x} = 1$
5. Therefore, $\ln L = 1$, and exponentiating both sides gives: $L = e$

Final Answer:

$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$

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Related Questions:

1. What is the significance of the constant $e$ in mathematics?
2. How can $e$ be derived using continuous compounding interest?
3. What is the natural logarithm and how does it relate to $e$?
4. How can this limit be proved using L'Hôpital's Rule?
5. What are other equivalent definitions of $e$ in calculus?

Tip: The constant $e$ plays a central role in calculus, especially in exponential growth, differential equations, and complex analysis.