https://file.aiquickdraw.com/mathbot/file/1725590088825yncwpywm.jpg

The given image shows the mathematical expression:

\[
\lim_{x \to 0} \frac{1}{x}
\]

This limit does **not exist**. Here's why:

- As \(x\) approaches \(0\) from the **positive side** (\(x \to 0^+\)), \(\frac{1}{x}\) approaches \(+\infty\), since dividing 1 by a very small positive number results in a very large positive value.
  
- As \(x\) approaches \(0\) from the **negative side** (\(x \to 0^-\)), \(\frac{1}{x}\) approaches \(-\infty\), since dividing 1 by a very small negative number results in a very large negative value.

Since the left-hand limit and right-hand limit are not equal, the two-sided limit does not exist.

Would you like more details or have any questions on this?

Here are five related questions to further explore limits:

1. What is the limit of \(\frac{1}{x}\) as \(x \to \infty\)?
2. How can we use L'Hôpital's Rule for indeterminate forms in limits?
3. What is the limit of \(x^2\) as \(x \to 0\)?
4. How does the concept of one-sided limits work in calculus?
5. Can you explain the difference between limits at infinity and limits at a point?

**Tip:** When evaluating limits, always check for behavior from both sides (left-hand and right-hand limits) if you suspect the limit might not exist!

Explore why the limit of 1/x as x approaches 0 does not exist. Understand the behavior of the function from both the positive and negative sides, and learn about one-sided limits. Suitable for high school calculus students.

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