problem is: Use continuity to evaluate the limit. \( \lim _{x \rightarrow 2} \arctan \left(\frac{x^{2}-4}{5 x^{2}-10 x}\right) \)

To evaluate the limit \[ \lim _{x \rightarrow 2} \arctan \left(\frac{x^{2}-4}{5 x^{2}-10 x}\right), \] we can start by simplifying the expression inside the arctan function. 1. **Simplify the expression**: The expression inside the limit is \[ \frac{x^{2}-4}{5 x^{2}-10 x}. \] Notice that \(x^2 - 4\) can be factored as \((x - 2)(x + 2)\) and \(5x^2 - 10x\) can be factored as \(5x(x - 2)\). Thus, we can rewrite the expression as: \[ \frac{(x - 2)(x + 2)}{5x(x - 2)}. \] 2. **Cancel common factors**: Since \(x \rightarrow 2\) and \(x - 2\) is a common factor, we can cancel it (as long as \(x \neq 2\)): \[ \frac{x + 2}{5x}. \] 3. **Evaluate the limit**: Now we can substitute \(x = 2\) into the simplified expression: \[ \lim_{x \rightarrow 2} \frac{x + 2}{5x} = \frac{2 + 2}{5 \cdot 2} = \frac{4}{10} = \frac{2}{5}. \] 4. **Apply continuity of arctan**: Since the arctan function is continuous, we can now evaluate the limit of the arctan function: \[ \lim_{x \rightarrow 2} \arctan \left(\frac{x + 2}{5x}\right) = \arctan\left(\frac{2}{5}\right). \] Thus, the final result is: \[ \lim _{x \rightarrow 2} \arctan \left(\frac{x^{2}-4}{5 x^{2}-10 x}\right) = \arctan\left(\frac{2}{5}\right). \]