b) \( \lim _{t \rightarrow 0} \frac{\tan (4 t)}{\tan (7 t)} \)

To find the limit \( \lim _{t \rightarrow 0} \frac{\tan (4 t)}{\tan (7 t)} \), we can use the small-angle approximation for the tangent function, which states that as \( t \) approaches 0, \( \tan(t) \approx t \). Applying this approximation: \[ \tan(4t) \approx 4t \quad \text{and} \quad \tan(7t) \approx 7t \] Substituting these into the limit gives: \[ \lim_{t \rightarrow 0} \frac{\tan(4t)}{\tan(7t)} \approx \lim_{t \rightarrow 0} \frac{4t}{7t} = \frac{4}{7} \] Thus, the limit evaluates to: \[ \frac{4}{7} \] So the final answer is: \[ \lim _{t \rightarrow 0} \frac{\tan (4 t)}{\tan (7 t)} = \frac{4}{7} \]