Examine `Lim_(xto 2)` [ x] , where [ x] denotes the greatest integar less than or equal to x.

Answer

To examine the limit \( \lim_{x \to 2} [x] \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we will evaluate both the right-hand limit and the left-hand limit.

### Step 1: Evaluate the Right-Hand Limit
The right-hand limit as \(x\) approaches 2 can be expressed as:
\[ \lim_{x \to 2^+} [x] = \lim_{h \to 0} [2 + h] \]
where \(h\) is a small positive number approaching 0.

As \(h\) approaches 0 from the right, \(2 + h\) will be slightly greater than 2. The greatest integer less than or equal to \(2 + h\) is:
\[ [2 + h] = 2 \]
Thus, we have:
\[ \lim_{x \to 2^+} [x] = 2 \]

### Step 2: Evaluate the Left-Hand Limit
Now, we evaluate the left-hand limit as \(x\) approaches 2:
\[ \lim_{x \to 2^-} [x] = \lim_{h \to 0} [2 - h] \]
where \(h\) is a small positive number approaching 0.

As \(h\) approaches 0 from the left, \(2 - h\) will be slightly less than 2. The greatest integer less than or equal to \(2 - h\) is:
\[ [2 - h] = 1 \]
Thus, we have:
\[ \lim_{x \to 2^-} [x] = 1 \]

### Step 3: Compare the Limits
Now we compare the right-hand limit and the left-hand limit:
- Right-hand limit: \( \lim_{x \to 2^+} [x] = 2 \)
- Left-hand limit: \( \lim_{x \to 2^-} [x] = 1 \)

Since the left-hand limit and the right-hand limit are not equal:
\[ \lim_{x \to 2^-} [x] \neq \lim_{x \to 2^+} [x] \]
we conclude that the limit does not exist.

### Final Conclusion
Thus, we can state:
\[ \lim_{x \to 2} [x] \text{ does not exist.} \]

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