To make the given function $f(x)$ continuous, we need to ensure that the left-hand limit as $x \to 0^-$, the right-hand limit as $x \to 0^+$, and the value at $x = 0$ all match. This involves the following three cases:

1. For $x < 0$, the function is:

$f(x) = \frac{28 \sin(x)}{x}$ As $x \to 0^-$, we need to evaluate the limit: $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{28 \sin(x)}{x}$ Using the standard limit result $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$, we get: $\lim_{x \to 0^-} f(x) = 28$

2. At $x = 0$, the function is $f(x) = a$. For continuity, we need:

$f(0) = a = 28$ Thus, $a = 28$.

3. For $x > 0$, the function is:

$f(x) = b \cos(x)$ As $x \to 0^+$, we need: $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} b \cos(x) = b \cdot \cos(0) = b \cdot 1 = b$ For continuity at $x = 0$, this must equal the value of the function at $x = 0$, which we already determined to be 28. Therefore, we have: $b = 28$

Conclusion:

To make the function continuous, the values of $a$ and $b$ must both be: $a = 28, \quad b = 28$

Would you like more details or have any questions?

Here are 5 related questions to expand this topic:

1. What is the general condition for continuity of piecewise functions?
2. How do limits play a role in ensuring the continuity of functions?
3. Can you find a scenario where $f(x)$ could be continuous but not differentiable at $x = 0$?
4. How would the function change if $b$ were dependent on $a$?
5. What other mathematical tools, like L'Hôpital's Rule, can help solve limit problems for continuity?

Tip: When dealing with piecewise functions, always check the limits from both sides to ensure continuity at the transition points.