Evaluate left-handed limit of the function : `f(x)={{:(abs(x-3)/(x-3)",",x ne 3), (" 0,", x=3):}` at x = 3.

Options

A. `1`
B. `-1`
C. `2`
D. `0`

Answer

To evaluate the left-handed limit of the function \( f(x) = \frac{|x-3|}{x-3} \) for \( x \neq 3 \) and \( f(x) = 0 \) for \( x = 3 \) at \( x = 3 \), we will follow these steps:

### Step 1: Understand the left-handed limit
The left-handed limit at \( x = 3 \) is denoted as \( \lim_{x \to 3^-} f(x) \). This means we are looking for the value of \( f(x) \) as \( x \) approaches 3 from the left (values less than 3).

**Hint:** Remember that the left-handed limit considers values approaching from the left side of the point.

### Step 2: Substitute into the limit expression
We need to evaluate:
\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} \frac{|x-3|}{x-3} \]

**Hint:** When substituting, keep in mind the behavior of the absolute value function.

### Step 3: Analyze the absolute value
For \( x < 3 \), the expression \( x - 3 \) is negative. Therefore, we have:
\[ |x-3| = -(x-3) = 3 - x \]

**Hint:** The absolute value function changes based on whether the argument is positive or negative.

### Step 4: Rewrite the limit
Now we can rewrite the limit:
\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} \frac{-(x-3)}{x-3} \]

**Hint:** Simplifying the expression will help in evaluating the limit.

### Step 5: Simplify the expression
The \( (x-3) \) terms in the numerator and denominator will cancel out (as long as \( x \neq 3 \)):
\[ \lim_{x \to 3^-} \frac{-(x-3)}{x-3} = \lim_{x \to 3^-} -1 = -1 \]

**Hint:** After canceling, check the remaining expression to find the limit.

### Step 6: Conclusion
Thus, the left-handed limit of the function \( f(x) \) as \( x \) approaches 3 is:
\[ \lim_{x \to 3^-} f(x) = -1 \]

**Final Answer:** The left-handed limit at \( x = 3 \) is \( -1 \).