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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.

Determine values of a and b that make the given function continuous. f(x) = 28sin(x)/x for x < 0, f(x) = a for x = 0, and f(x) = bcos(x) for x > 0.

Learn how to determine the values of a and b that make the given piecewise function continuous. The function involves trigonometric expressions such as 28sin(x)/x and bcos(x), and this solution covers limits, continuity, and trigonometric limits in detail.

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