Questions: Question 3 of 8, Step 3 of 3 4 / 12 Correct Consider the function f(x)= x^2+8x+22 if x<-4 2x+14 if x ≥ -4 Step 3 of 3: Find the limit as x approaches -4 of f(x).

Solution Steps

To find the limit of the piecewise function \( f(x) \) as \( x \) approaches \(-4\), we need to evaluate the left-hand limit and the right-hand limit separately. If both limits are equal, then the limit exists and is equal to that common value.

1. Evaluate the left-hand limit as \( x \) approaches \(-4\) from the left (\( x < -4 \)).
2. Evaluate the right-hand limit as \( x \) approaches \(-4\) from the right (\( x \geq -4 \)).
3. Compare the two limits. If they are equal, that is the limit of \( f(x) \) as \( x \) approaches \(-4\).

The function \( f(x) \) is defined as follows: \[ f(x) = \begin{cases} x^2 + 8x + 22 & \text{if } x < -4 \\ 2x + 14 & \text{if } x \geq -4 \end{cases} \]

To find the left-hand limit as \( x \) approaches \(-4\), we use the first piece of the function: \[ \lim_{x \to -4^-} f(x) = (-4)^2 + 8(-4) + 22 = 16 - 32 + 22 = 6 \]

To find the right-hand limit as \( x \) approaches \(-4\), we use the second piece of the function: \[ \lim_{x \to -4^+} f(x) = 2(-4) + 14 = -8 + 14 = 6 \]

Since both the left-hand limit and the right-hand limit are equal: \[ \lim_{x \to -4^-} f(x) = 6 \quad \text{and} \quad \lim_{x \to -4^+} f(x) = 6 \] we conclude that: \[ \lim_{x \to -4} f(x) = 6 \]

Final Answer