Show that `Lim_(x to 0 ) 1/x ` does not exist .

Answer

To show that \(\lim_{x \to 0} \frac{1}{x}\) does not exist, we will evaluate the right-hand limit and the left-hand limit separately.

### Step 1: Evaluate the Right-Hand Limit

We start by calculating the right-hand limit:

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

This means we are approaching 0 from the positive side. As \(x\) gets closer to 0 from the right (positive values), \(\frac{1}{x}\) becomes larger and larger. Mathematically, we can express this as:

\[ \lim_{x \to 0^+} \frac{1}{x} = \lim_{h \to 0^+} \frac{1}{h} = +\infty \]

### Step 2: Evaluate the Left-Hand Limit

Next, we calculate the left-hand limit:

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

This means we are approaching 0 from the negative side. As \(x\) gets closer to 0 from the left (negative values), \(\frac{1}{x}\) becomes more and more negative. We can express this as:

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

### Step 3: Compare the Limits

Now we compare the right-hand limit and the left-hand limit:

- Right-hand limit: \(\lim_{x \to 0^+} \frac{1}{x} = +\infty\)
- Left-hand limit: \(\lim_{x \to 0^-} \frac{1}{x} = -\infty\)

Since the right-hand limit and the left-hand limit are not equal (one is \(+\infty\) and the other is \(-\infty\)), we conclude that the overall limit does not exist.

### Conclusion

Thus, we can say:

\[ \lim_{x \to 0} \frac{1}{x} \text{ does not exist.} \]