Let `f(x) = {((x^(2) -2x -3)/(x + 1)","," when " x != -1),(" k,"," when " x = -1):}` If f(x) is continuous at `x = -1` then k = ?

Options

A. 4
B. `-4`
C. `-3`
D. 2

Explanation

To determine the value of \( k \) such that the function \( f(x) \) is continuous at \( x = -1 \), we need to follow these steps:

### Step 1: Understand the definition of continuity
A function \( f(x) \) is continuous at a point \( c \) if:
\[ \lim_{x \to c} f(x) = f(c) \]
In this case, we need to check the continuity at \( x = -1 \).

### Step 2: Find the limit of \( f(x) \) as \( x \) approaches -1
Given:
\[ f(x) = \frac{x^2 - 2x - 3}{x + 1} \quad \text{for } x \neq -1 \]
We need to calculate:
\[ \lim_{x \to -1} f(x) = \lim_{x \to -1} \frac{x^2 - 2x - 3}{x + 1} \]

### Step 3: Factor the numerator
First, we factor the numerator \( x^2 - 2x - 3 \):
\[ x^2 - 2x - 3 = (x + 1)(x - 3) \]
Thus, we can rewrite the function:
\[ f(x) = \frac{(x + 1)(x - 3)}{x + 1} \quad \text{for } x \neq -1 \]

### Step 4: Simplify the function
For \( x \neq -1 \), we can cancel \( (x + 1) \):
\[ f(x) = x - 3 \quad \text{for } x \neq -1 \]

### Step 5: Calculate the limit
Now we can find the limit as \( x \) approaches -1:
\[ \lim_{x \to -1} f(x) = \lim_{x \to -1} (x - 3) = -1 - 3 = -4 \]

### Step 6: Set the limit equal to \( k \)
For \( f(x) \) to be continuous at \( x = -1 \), we need:
\[ f(-1) = k = \lim_{x \to -1} f(x) = -4 \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{-4} \]

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