Limits and Continuity

(cod: P-45-20-8) Determine the values of the constants \(a\) and \(b\) so that the function below is continuous at \(x=0\): \[f(x) = \begin{cases} \dfrac{\sin{(2x)}}{ax}, \text{ if } x<0, \\ \\ \dfrac{\sqrt{x^2+4} - \sqrt{x+4}}{2x}, \text{ if } x>0,\\ \\ b, \text{ if } x=0. \end{cases}\]

\(a = - 16 \ \ \ \) and \( \ \ \ b = - \dfrac{1}{8}.\)

\(a = - 16 \ \ \ \) and \( \ \ \ b = \dfrac{1}{8}.\)

\(a = 8 \ \ \ \) and \( \ \ \ b = - \dfrac{1}{8}.\)

\(a = - 8 \ \ \ \) and \( \ \ \ b = - \dfrac{1}{8}.\)

\(a = 16 \ \ \ \) and \( \ \ \ b = - \dfrac{1}{8}.\)