The problem in the image involves finding the limit of the expression as $x \to 0$. The given expression is:

$\lim_{x \to 0} \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}$

Let's solve this step by step.

Step 1: Rewrite the expression

The expression is:

$f(x) = \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}$

Step 2: Analyze the limit as $x \to 0$

When $x \to 0$, the terms $\frac{1}{x}$ tend to either $+\infty$ or $-\infty$ depending on the direction of approach (from positive or negative values of $x$).

• If $x \to 0^+$ (approaching 0 from the right, where $x > 0$):

  • $\frac{1}{x} \to +\infty$
  • The expression becomes $\frac{1 + \infty}{1 - \infty} = \frac{\infty}{-\infty}$, which is undefined.

• If $x \to 0^-$ (approaching 0 from the left, where $x < 0$):

  • $\frac{1}{x} \to -\infty$
  • The expression becomes $\frac{1 - \infty}{1 + \infty} = \frac{-\infty}{\infty}$, which is again undefined.

Step 3: Conclusion

Since the limit behavior is different from both the left and the right, the limit does not exist at $x = 0$.

Would you like a more detailed explanation of the steps or analysis? Here are some related questions you can explore:

1. What are limits and when do they not exist?
2. How do one-sided limits work in cases like this?
3. Can L'Hopital's Rule be applied here, and why or why not?
4. What happens if we modify the expression slightly to make the limit exist?
5. How can limits involving infinity be interpreted in practical terms?

Tip: Always check the behavior of a limit from both sides to confirm whether the limit exists or not, especially when the function involves infinity.