`lim_(xrarre) (log_(e)x-1)/(|x-e|)` is

Options

A. `(1)/(e)`
B. `-(1)/(e)`
C. e
D. none -existent

Explanation

To solve the limit \( \lim_{x \to e} \frac{\log_e x - 1}{|x - e|} \), we will analyze the limit from both the right-hand side and the left-hand side.

### Step 1: Right-Hand Limit
We first calculate the right-hand limit as \( x \) approaches \( e \) from the right (i.e., \( x \to e^+ \)).

\[ \lim_{x \to e^+} \frac{\log_e x - 1}{|x - e|} = \lim_{x \to e^+} \frac{\log_e x - 1}{x - e} \]

Since \( x \) is approaching \( e \) from the right, \( |x - e| = x - e \).

### Step 2: Evaluate the Limit
As \( x \to e \), both the numerator and the denominator approach 0, resulting in the indeterminate form \( \frac{0}{0} \). We can apply L'Hôpital's Rule:

\[ \text{Differentiate the numerator: } \frac{d}{dx}(\log_e x - 1) = \frac{1}{x} \]
\[ \text{Differentiate the denominator: } \frac{d}{dx}(x - e) = 1 \]

Now, applying L'Hôpital's Rule:

\[ \lim_{x \to e^+} \frac{\log_e x - 1}{x - e} = \lim_{x \to e^+} \frac{\frac{1}{x}}{1} = \frac{1}{e} \]

### Step 3: Left-Hand Limit
Next, we calculate the left-hand limit as \( x \) approaches \( e \) from the left (i.e., \( x \to e^- \)).

\[ \lim_{x \to e^-} \frac{\log_e x - 1}{|x - e|} = \lim_{x \to e^-} \frac{\log_e x - 1}{e - x} \]

Since \( x \) is approaching \( e \) from the left, \( |x - e| = e - x \).

### Step 4: Evaluate the Left-Hand Limit
Again, we have the indeterminate form \( \frac{0}{0} \) as \( x \to e \). We apply L'Hôpital's Rule again:

\[ \text{Differentiate the numerator: } \frac{d}{dx}(\log_e x - 1) = \frac{1}{x} \]
\[ \text{Differentiate the denominator: } \frac{d}{dx}(e - x) = -1 \]

Now, applying L'Hôpital's Rule:

\[ \lim_{x \to e^-} \frac{\log_e x - 1}{e - x} = \lim_{x \to e^-} \frac{\frac{1}{x}}{-1} = -\frac{1}{e} \]

### Step 5: Conclusion
Now we compare the right-hand limit and the left-hand limit:

- Right-hand limit: \( \frac{1}{e} \)
- Left-hand limit: \( -\frac{1}{e} \)

Since the two limits are not equal, the limit does not exist.

Thus, the final answer is:

\[ \text{The limit does not exist.} \]