Give an example of a function f in which 0 is in the Domaine(f) but the limit of function (F) is undefined

An example of a function \( f(x) \) where 0 is in the domain but the limit as \( x \) approaches 0 is undefined is the function:

\[
f(x) = \frac{1}{x}
\]

### Explanation:

- **Domain of \( f(x) \)**: The function \( f(x) = \frac{1}{x} \) is defined for all real numbers except \( x = 0 \). However, you can define \( f(0) \) as any value (say 0), meaning 0 is in the domain of \( f \).
  
  For example:
  \[
  f(x) = 
  \begin{cases}
  \frac{1}{x}, & \text{if } x \neq 0 \\
  0, & \text{if } x = 0
  \end{cases}
  \]

- **Limit as \( x \to 0 \)**: The limit of \( f(x) \) as \( x \to 0 \) does not exist. This is because:

  \[
  \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{1}{x} = +\infty
  \]
  \[
  \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{1}{x} = -\infty
  \]

  Since the left-hand limit and the right-hand limit are different, the overall limit does not exist.

### Conclusion:
Even though 0 is in the domain of the function, the limit of \( f(x) \) as \( x \to 0 \) is undefined.

Do you want further clarification or have any questions?

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Here are 5 related questions:
1. What happens if you define \( f(0) \) differently?
2. Can a function have a limit at a point even if the function is not defined at that point?
3. What is the definition of one-sided limits?
4. How do we define limits when they tend toward infinity?
5. What are some real-world examples of functions with undefined limits?

**Tip**: Always check both one-sided limits when determining whether the overall limit exists at a point.

Give an example of a function f in which 0 is in the Domaine(f) but the limit of function (F) is undefined.

Explore an example of a function where 0 is in the domain, but the limit at 0 is undefined. Learn about one-sided limits and why the limit of f(x) = 1/x does not exist as x approaches 0 from both sides. Suitable for advanced high school students studying calculus.

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