Question

The value of $\displaystyle \lim_{x \to \infty}\left(1+\frac{2}{3x}\right)^{x}$ is:

• A. $e$
• B. $e^{2}$
• C. $e^{\tfrac{2}{3}}$
• D. $\dfrac{1}{e^{3}}$

Hint:

Memorize $\left(1+\tfrac{k}{x}\right)^{x}\!\to e^{k}$; it converts many compound-interest–type limits directly to $e^{k}$.

Answer 1

The Correct Option is C

Step 1: Use the standard exponential limit.
Recall $\displaystyle \lim_{x\to\infty}\left(1+\frac{k}{x}\right)^{x}=e^{k}$.

Step 2: Match the form.
Here $\left(1+\dfrac{2}{3x}\right)^{x}=\left(1+\dfrac{\tfrac{2}{3}}{x}\right)^{x}$, so $k=\tfrac{2}{3}$.

Step 3: Evaluate the limit.
Therefore $\displaystyle \lim_{x\to\infty}\left(1+\dfrac{\tfrac{2}{3}}{x}\right)^{x}=e^{\tfrac{2}{3}}$.

Step 4: Conclusion.
The required value is $e^{\tfrac{2}{3}}$.