Consider a function f:R→R*f*:R→R defined as f(x)={sin⁡17x+Asin⁡x15x3if x≠0,Bif x=0.*f*(*x*)={15*x*3sin17*x*+*A*sin*x*​*B*​if *x*=0,if *x*=0.​ If f(x)*f*(*x*) is continuous at x=0*x*=0 , then find the value of 90B−A90*B*−*A*

To determine the value of \( 90B - A \), we need to ensure that the function \( f(x) \) is continuous at \( x = 0 \).

The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} 
\sin(17x) + A \sin(x) \frac{1}{15x^3} & \text{if } x \neq 0 \\
B & \text{if } x = 0
\end{cases} \]

For \( f(x) \) to be continuous at \( x = 0 \), the following condition must be met:
\[ \lim_{x \to 0} f(x) = f(0) \]

That is:
\[ \lim_{x \to 0} \left( \sin(17x) + A \sin(x) \frac{1}{15x^3} \right) = B \]

First, we evaluate the limit of the first term as \( x \to 0 \):
\[ \lim_{x \to 0} \sin(17x) = \sin(0) = 0 \]

Next, we examine the second term:
\[ \lim_{x \to 0} A \sin(x) \frac{1}{15x^3} \]

Since \( \sin(x) \approx x \) when \( x \) is close to 0, we can approximate:
\[ \sin(x) \approx x \]

Thus, the second term becomes:
\[ A \frac{x}{15x^3} = \frac{A}{15x^2} \]

So we need:
\[ \lim_{x \to 0} \left( 0 + \frac{A}{15x^2} \right) = B \]

We observe that:
\[ \frac{A}{15x^2} \to \infty \text{ as } x \to 0 \text{ unless } A = 0 \]

For the limit to be finite (i.e., equal to \( B \)), it must be the case that \( A = 0 \). Hence:
\[ \lim_{x \to 0} \left( \sin(17x) + 0 \right) = B \]

This simplifies to:
\[ \lim_{x \to 0} \sin(17x) = B \]

Since:
\[ \lim_{x \to 0} \sin(17x) = \sin(0) = 0 \]

It follows that:
\[ B = 0 \]

Finally, substituting \( A = 0 \) and \( B = 0 \) into the expression \( 90B - A \), we get:
\[ 90B - A = 90(0) - 0 = 0 \]

Therefore, the value of \( 90B - A \) is:
\[ \boxed{0} \]

Would you like more details on any part of this solution, or do you have any questions? 

Here are some related questions to further explore the topic:

1. How do we rigorously prove the limit of a function?
2. What are the properties of the sine function near zero?
3. Why is it important to consider the limit when ensuring continuity?
4. How does the small-angle approximation for sine work?
5. What happens to other trigonometric functions near zero?
6. Can you generalize this problem for other trigonometric functions?
7. What are the implications of having \( A \neq 0 \) in such functions?
8. How do you handle continuity conditions for piecewise functions in general?

**Tip:** When evaluating limits, especially for piecewise functions, always carefully consider the behavior of each piece separately and ensure that they match at the points where they join.

Learn how to solve the problem of ensuring continuity of a piecewise defined function involving trigonometric functions. This problem covers limits, trigonometric functions near zero, and the small-angle approximation for sine. Suitable for college-level students.

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