Use continuity to evaluate the limit. \( \lim _{x \rightarrow 4} \arctan \left(\frac{x^{2}-16}{3 x^{2}-12 x}\right) \)

To evaluate the limit \[ \lim _{x \rightarrow 4} \arctan \left(\frac{x^{2}-16}{3 x^{2}-12 x}\right), \] let's first simplify the expression inside the arctangent function. We start with the numerator and denominator: 1. **Numerator:** \( x^2 - 16 = (x-4)(x+4) \) 2. **Denominator:** \[ 3x^2 - 12x = 3(x^2 - 4x) = 3x(x - 4). \] Now, we can rewrite the limit: \[ \frac{x^2 - 16}{3x^2 - 12x} = \frac{(x-4)(x+4)}{3x(x-4)}. \] For \( x \neq 4 \), we can cancel out \( (x-4) \): \[ \frac{x+4}{3x}. \] So we have: \[ \lim_{x \rightarrow 4} \arctan\left(\frac{x+4}{3x}\right). \] Now, we can directly substitute \( x = 4 \): \[ \frac{4+4}{3 \cdot 4} = \frac{8}{12} = \frac{2}{3}. \] Therefore, we have: \[ \lim_{x \rightarrow 4} \arctan\left(\frac{x+4}{3x}\right) = \arctan\left(\frac{2}{3}\right). \] Thus, the evaluated limit is: \[ \boxed{\arctan\left(\frac{2}{3}\right)}. \]