To find the limit [tex]\(\lim_{x \rightarrow 0} \frac{\sin(a x)}{x \cos(\beta x)}\)[/tex], we can approach it step-by-step:
1. Identify the expression:
[tex]\[
\frac{\sin(a x)}{x \cos(\beta x)}
\][/tex]
2. Analyze the expressions for small values of [tex]\( x \)[/tex]:
- As [tex]\( x \)[/tex] approaches 0, [tex]\(\sin(a x) \)[/tex] closely approximates [tex]\(a x \)[/tex]. This is based on the well-known limit:
[tex]\[
\lim_{x \to 0} \frac{\sin(a x)}{a x} = 1
\][/tex]
- Similarly, as [tex]\( x \)[/tex] approaches 0, [tex]\(\cos(\beta x) \)[/tex] approximates to 1. This is because:
[tex]\[
\cos(0) = 1
\][/tex]
3. Substitute these approximations into the limit expression:
[tex]\[
\frac{\sin(a x)}{x \cos(\beta x)} \approx \frac{a x}{x \cdot 1}
\][/tex]
4. Simplify the expression:
[tex]\[
\frac{a x}{x} = a
\][/tex]
5. Taking the limit:
- Since the simplified expression is constant with respect to [tex]\( x \)[/tex]:
[tex]\[
\lim_{x \rightarrow 0} a = a
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{a}
\][/tex]
