Question

The value of $\lim_{x\rightarrow\infty}(1 + \frac{1}{x})^{x} = $

• A. 0
• B. 1
• C. $e$
• D. $\infty$

Hint:

Memorize this! $(1 + 1/x)^x \to e$ as $x \to \infty$. It's the basis for continuous compounding and many calculus proofs.

Answer 1

The Correct Option is C

Step 1: Concept
This is the fundamental limit definition of the mathematical constant $e$.

Step 2: Meaning
As $x$ approaches infinity, the expression $(1 + 1/x)^{x}$ approaches a specific irrational number.

Step 3: Analysis
Using the formula $\lim_{x\rightarrow a} [f(x)]^{g(x)} = e^{\lim_{x\rightarrow a} g(x)[f(x)-1]}$, we get $e^{\lim_{x\rightarrow \infty} x[1 + 1/x - 1]} = e^{\lim_{x\rightarrow \infty} x(1/x)} = e^{1}$.

Step 4: Conclusion
The limit is $e$.
Final Answer: (C)