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To make the function \( f(x) \) continuous at \( x = -7 \), the left-hand limit (i.e., the limit of \( f(x) \) as \( x \) approaches \(-7\)) must equal \( f(-7) = k \).

The function \( f(x) \) is defined as:

\[
f(x) = 
\begin{cases} 
\frac{x^2 + x - 20}{x^2 + 3x - 28} & \text{for} \ x \neq -7 \\
k & \text{for} \ x = -7
\end{cases}
\]

### Step 1: Factorize the function's expression
We begin by factoring both the numerator and the denominator:

\[
x^2 + x - 20 = (x - 4)(x + 5)
\]
\[
x^2 + 3x - 28 = (x - 4)(x + 7)
\]

So the expression for \( f(x) \) when \( x \neq -7 \) becomes:

\[
f(x) = \frac{(x - 4)(x + 5)}{(x - 4)(x + 7)} \quad \text{for} \ x \neq -7
\]

For \( x \neq -4 \), the \( (x - 4) \) terms cancel out, leaving:

\[
f(x) = \frac{x + 5}{x + 7} \quad \text{for} \ x \neq -7
\]

### Step 2: Find the limit as \( x \to -7 \)
To ensure continuity at \( x = -7 \), we need to compute the limit of \( f(x) \) as \( x \to -7 \):

\[
\lim_{x \to -7} \frac{x + 5}{x + 7}
\]

Substituting \( x = -7 \) into this expression:

\[
\lim_{x \to -7} \frac{x + 5}{x + 7} = \frac{-7 + 5}{-7 + 7} = \frac{-2}{0}
\]

This results in an undefined expression, suggesting a vertical asymptote at \( x = -7 \). Therefore, there is no value of \( k \) that can make the function continuous at \( x = -7 \).

Thus, the function cannot be made continuous at \( x = -7 \) for any value of \( k \).

Would you like a more detailed explanation or have any questions?

Here are some follow-up questions:

1. What is the general definition of continuity for a function at a point?
2. Why is the function undefined at \( x = -7 \) when calculating the limit?
3. Can you explain what a vertical asymptote is and how it affects continuity?
4. How would this problem change if the function's denominator didn't vanish at \( x = -7 \)?
5. What is the significance of factoring rational functions in problems involving limits?

**Tip:** Always check for removable discontinuities by simplifying the expression before calculating limits.

Find all values of k that would make the function f(x) continuous at x = -7.

This problem explores how to find the value of k that makes a rational function continuous at a given point. Key concepts include limits, rational functions, and factorization. Suitable for Grades 11-12.

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