Questions: Determine the value for c so that the limit as x approaches 4 of f(x) exists. f(x) = (1/4) x + c, for x < 4 -x + 7, for x > 4 The value of c is.

Transcript text: Determine the value for $c$ so that $\lim _{x \rightarrow 4} f(x)$ exists. \[ f(x)=\left\{\begin{array}{ll} \frac{1}{4} x+c, & \text { for } x<4 \\ -x+7, & \text { for } x>4 \end{array}\right. \] The value of $c$ is $\square$

Solution

Solution Steps

Step 1: Understand the problem

We are given a piecewise function $ f(x) $ defined as: \[ f(x)=\left\{\begin{array}{ll} \frac{1}{4} x+c, & \text { for } x<4 \\ -x+7, & \text { for } x>4 \end{array}\right. \] We need to determine the value of $ c $ such that the limit $ \lim _{x \rightarrow 4} f(x) $ exists.

Step 2: Determine the left-hand limit

The left-hand limit as $ x $ approaches 4 is given by the expression for $ x < 4 $: \[ \lim_{x \rightarrow 4^-} f(x) = \frac{1}{4} \cdot 4 + c = 1 + c. \]

Step 3: Determine the right-hand limit

The right-hand limit as $ x $ approaches 4 is given by the expression for $ x > 4 $: \[ \lim_{x \rightarrow 4^+} f(x) = -4 + 7 = 3. \]

Step 4: Set the left-hand limit equal to the right-hand limit

For the limit $ \lim_{x \rightarrow 4} f(x) $ to exist, the left-hand limit and the right-hand limit must be equal: \[ 1 + c = 3. \]

Step 5: Solve for $ c $

Solving the equation $ 1 + c = 3 $ gives: \[ c = 3 - 1 = 2. \]