From the provided image, the expression under consideration is:

$\lim_{x \to +\infty} \left(1 - \frac{2}{x}\right)^x$

This is a limit problem involving an exponential expression. The steps to evaluate this limit are as follows:

Step 1: Recognize the Indeterminate Form

As $x \to +\infty$, the base $1 - \frac{2}{x} \to 1$, and the exponent $x \to \infty$. Hence, the expression is of the indeterminate form $1^\infty$.

Step 2: Rewrite Using Logarithms

We rewrite the expression by setting it as $e^{\ln[(1 - \frac{2}{x})^x]}$: $\lim_{x \to +\infty} \left(1 - \frac{2}{x}\right)^x = \lim_{x \to +\infty} e^{x \ln\left(1 - \frac{2}{x}\right)}$

Step 3: Simplify the Logarithmic Term

The natural logarithm $\ln(1 - \frac{2}{x})$ can be approximated for large $x$ using the expansion $\ln(1 + u) \approx u$ when $u$ is small. Here, $u = -\frac{2}{x}$, so: $\ln\left(1 - \frac{2}{x}\right) \approx -\frac{2}{x}.$

Substitute this back: $x \ln\left(1 - \frac{2}{x}\right) \approx x \left(-\frac{2}{x}\right) = -2.$

Step 4: Evaluate the Exponential Limit

Thus: $\lim_{x \to +\infty} \left(1 - \frac{2}{x}\right)^x = \lim_{x \to +\infty} e^{-2} = e^{-2}.$

Final Answer:

$\boxed{e^{-2}}$

Would you like further explanation, or should I clarify a specific step?
Here are five questions to deepen your understanding:

1. Why does $\ln(1 - \frac{2}{x}) \to -\frac{2}{x}$ for large $x$?
2. How does recognizing $1^\infty$ as an indeterminate form guide the solution process?
3. What is the significance of using logarithmic transformations in exponential limits?
4. How can the series expansion of $\ln(1 + u)$ be formally derived?
5. What are other examples of limits involving indeterminate forms like $1^\infty$?

Tip: For any indeterminate limit of the form $1^\infty$, rewriting the base using the natural logarithm is a standard and reliable method.