ABADEMCS SUPRORT \[ \begin{array}{l}\text { Use the Squeeze Theorem to find } \lim _{x \rightarrow 4} f(x) \text {, given that } 7-|x-4| \leq f(x) \leq 7+|x-4| \\ \lim _{x \rightarrow 4} f(x)=\text { Number }\end{array} \]

We are given that \[ 7 - |x-4| \leq f(x) \leq 7 + |x-4| \] and we need to find \[ \lim_{x \rightarrow 4} f(x). \] **Step 1: Evaluate the limits of the bounding functions** Consider the lower bound: \[ \lim_{x \rightarrow 4} \left(7 - |x-4|\right) \] Since \(|x-4|\) becomes \(0\) as \(x\) approaches \(4\), we have: \[ \lim_{x \rightarrow 4} \left(7 - |x-4|\right) = 7 - 0 = 7. \] Similarly, for the upper bound: \[ \lim_{x \rightarrow 4} \left(7 + |x-4|\right) \] we have: \[ \lim_{x \rightarrow 4} \left(7 + |x-4|\right) = 7 + 0 = 7. \] **Step 2: Apply the Squeeze Theorem** Since \[ 7 - |x-4| \leq f(x) \leq 7 + |x-4| \] and both the lower and upper bounds have the same limit at \(x \rightarrow 4\), by the Squeeze Theorem: \[ \lim_{x \rightarrow 4} f(x) = 7. \] **Final Answer** \[ \lim_{x \rightarrow 4} f(x) = 7. \]