Let's analyze each part step-by-step using the graph provided. --- ### (a) \(\lim_{x \to 1^-} f(x)\) - This is the left-hand limit as \(x\) approaches 1. - As \(x\) approaches 1 from the left, the graph **shoots upward toward \(+\infty\)**. - **Answer:** \(\boxed{+\infty}\) --- ### (b) \(\lim_{x \to 3^+} f(x)\) - This is the right-hand limit as \(x\) approaches 3. - As \(x\) approaches 3 from the right, the graph **approaches 1** (the value at the solid dot just right of \(x = 3\)). - **Answer:** \(\boxed{1}\) --- ### (c) \(\lim_{x \to 3} f(x)\) - For the two-sided limit, both left and right limits must agree. - Left of \(x = 3\), the graph approaches **4** (solid dot at \(x = 3\) from the left). - Right of \(x = 3\), the graph approaches **1** (solid dot at \(x = 3\) from the right). - Since \(4 \neq 1\), the limit **does not exist**. - **Answer:** \(\boxed{\text{DNE}}\) --- ### (d) \(f(3)\) - The value of the function at \(x = 3\) is at the **open circle** at \(y = -2\). - **Answer:** \(\boxed{-2}\) --- ### (e) \(f'(2)\) - The derivative at \(x = 2\) is the **slope** of the graph at that point. - From \(x = 1\) to \(x = 3\), the graph is a straight line. - The line passes through points \((1, 0)\) and \((3, 4)\). - Slope = \(\frac{4 - 0}{3 - 1} = 2\). - **Answer:** \(\boxed{2}\) --- ### (f) List all \(x\)-values where \(f\) is **discontinuous**. - Discontinuities occur at: - \(x = 1\) (vertical asymptote) - \(x = 3\) (jump discontinuity: left and right limits differ, and the function value is different from both) - \(x = 6\) (open circle at \(y = 1\), function jumps to \(y = 3\)) - **Answer:** \(\boxed{1, 3, 6}\) --- ## **Summary of Answers** (a) \(+\infty\) (b) \(1\) (c) \(\text{DNE}\) (d) \(-2\) (e) \(2\) (f) \(x = 1, 3, 6\)