We are tasked with finding the left and right sided limits of the function:

$f(x) = \frac{1}{1 + 2 - \frac{1}{x}} \quad \text{as} \quad x \to 0.$

Step 1: Simplifying the function

Let's first simplify the expression for $f(x)$:

$f(x) = \frac{1}{1 + 2 - \frac{1}{x}} = \frac{1}{3 - \frac{1}{x}}.$

Step 2: Analyzing the Left-Hand Limit ($x \to 0^-$)

For $x \to 0^-$ (approaching zero from the left), $x$ is negative. Therefore, $\frac{1}{x}$ will be a large negative value.

$\frac{1}{x} \to -\infty \quad \text{as} \quad x \to 0^-.$

Thus, the expression for $f(x)$ becomes:

$f(x) = \frac{1}{3 - \left( -\infty \right)} = \frac{1}{3 + \infty} = \frac{1}{\infty} = 0.$

Hence, the left-hand limit is:

$\lim_{x \to 0^-} f(x) = 0.$

Step 3: Analyzing the Right-Hand Limit ($x \to 0^+$)

For $x \to 0^+$ (approaching zero from the right), $x$ is positive. Therefore, $\frac{1}{x}$ will be a large positive value.

$\frac{1}{x} \to +\infty \quad \text{as} \quad x \to 0^+.$

Thus, the expression for $f(x)$ becomes:

$f(x) = \frac{1}{3 - \left( +\infty \right)} = \frac{1}{3 - \infty} = \frac{1}{-\infty} = 0.$

Hence, the right-hand limit is:

$\lim_{x \to 0^+} f(x) = 0.$

Conclusion:

Both the left-hand and right-hand limits of the function $f(x) = \frac{1}{3 - \frac{1}{x}}$ as $x \to 0$ are equal to 0:

$\lim_{x \to 0^-} f(x) = 0 \quad \text{and} \quad \lim_{x \to 0^+} f(x) = 0.$

Would you like further explanation or have additional questions? Here are some related questions you might find interesting:

1. What happens to the function when $x$ approaches other values, like $x \to \infty$?
2. How does the function behave as $x \to -\infty$?
3. What are the conditions for a function to have a limit at a specific point?
4. How would the behavior of this function change if the denominator had a different form, such as $3 + \frac{1}{x}$?
5. Can this function be classified as continuous at $x = 0$?

Tip: When calculating limits of rational functions as $x$ approaches infinity or zero, always look at the behavior of terms with $x$ in the denominator, as they often dominate the function’s behavior.