If `f(x)=(1)/(1-|1-x|)` then what is `lim_(xto1)f(x)` equal to ?

Options

A. 0
B. `oo`
C. 1
D. `-1`

Answer

To find the limit of the function \( f(x) = \frac{1}{1 - |1 - x|} \) as \( x \) approaches 1, we will follow these steps:

### Step 1: Understand the function
The function is defined as:
\[ f(x) = \frac{1}{1 - |1 - x|} \]
We need to evaluate the limit as \( x \) approaches 1.

### Step 2: Analyze \( |1 - x| \)
As \( x \) approaches 1, we need to consider the expression \( |1 - x| \).

- If \( x < 1 \), then \( 1 - x > 0 \) and \( |1 - x| = 1 - x \).
- If \( x = 1 \), then \( |1 - x| = 0 \).
- If \( x > 1 \), then \( 1 - x < 0 \) and \( |1 - x| = x - 1 \).

Since we are taking the limit as \( x \) approaches 1, we will consider the left-hand limit (as \( x \) approaches 1 from the left).

### Step 3: Substitute \( x \) in the function
For \( x \) approaching 1 from the left:
\[ |1 - x| = 1 - x \]
Thus, we can rewrite the function:
\[ f(x) = \frac{1}{1 - (1 - x)} = \frac{1}{x} \]

### Step 4: Evaluate the limit
Now we will evaluate the limit:
\[ \lim_{x \to 1} f(x) = \lim_{x \to 1} \frac{1}{x} \]
Substituting \( x = 1 \):
\[ = \frac{1}{1} = 1 \]

### Conclusion
Thus, the limit is:
\[ \lim_{x \to 1} f(x) = 1 \]

### Final Answer
The limit \( \lim_{x \to 1} f(x) \) is equal to \( 1 \).

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