Let ​f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column x squared plus 7 comma 2nd Column x less than minus 7 2nd Row 1st Column StartRoot x plus 7 EndRoot comma 2nd Column x greater than or equals minus 7 EndMatrix
Compute the limit of ModifyingBelow lim With x right arrow minus 7 Superscript minus f left parenthesis x right parenthesis or state that it does not exist. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

Let's compute the limit of the function \( f(x) \) as \( x \) approaches \( -7 \) from the left, denoted as \( \lim_{x \to -7^-} f(x) \).

### Given Function

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

\[
f(x) = 
\begin{cases} 
x^2 + 7, & \text{if } x < -7 \\
\sqrt{x + 7}, & \text{if } x \geq -7 
\end{cases}
\]

### Problem: Compute \( \lim_{x \to -7^-} f(x) \)

We are interested in the limit as \( x \) approaches \( -7 \) from the **left** side. This means we will use the expression for \( f(x) \) where \( x < -7 \).

#### Step 1: Find the Limit Using the Appropriate Piece

Since \( x \) is approaching \( -7 \) from the left (\( x \to -7^- \)), we use the first piece of the function:

\[
f(x) = x^2 + 7 \quad \text{for } x < -7
\]

#### Step 2: Calculate the Limit

Now, compute the limit:

\[
\lim_{x \to -7^-} (x^2 + 7)
\]

Substitute \( x = -7 \):

\[
(-7)^2 + 7 = 49 + 7 = 56
\]

### Conclusion

\[
\lim_{x \to -7^-} f(x) = 56
\]

### Answer

The correct choice is:

\[
\text{The limit is } 56.
\]

Would you like more details on how we handled this piecewise function or more examples?

#### Related Questions:
1. What is the right-hand limit of \( f(x) \) as \( x \to -7^+ \)?
2. How do you determine if a piecewise function is continuous at a point?
3. How do you compute limits involving square roots?
4. What happens if a limit does not exist?
5. How can we graphically interpret piecewise functions to understand their limits?

**Tip:** Always consider the direction of the limit (from the left or the right) when dealing with piecewise functions, as different expressions may apply!

Learn how to compute the limit of a piecewise function as x approaches -7. This problem involves analyzing limits and piecewise functions, specifically focusing on a function defined by different rules for x less than -7 and x greater than or equal to -7. Suitable for advanced high school students.

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