To solve the limit [tex]\(\lim _{x \rightarrow 0} \frac{\sin(8x)}{\sin(4x)}\)[/tex], follow these steps:
1. Expression Observation:
Notice that both the numerator and the denominator involve trigonometric sine functions that are multiplied by constants.
2. Basic Trigonometric Limit Use:
Recall the fundamental trigonometric limit:
[tex]\[
\lim_{x \to 0} \frac{\sin(kx)}{kx} = 1
\][/tex]
for any constant [tex]\( k \)[/tex].
3. Rewriting Using a Substitution:
We can manipulate the given limit to leverage this basic limit. Let's rewrite the given limit:
[tex]\[
\lim_{x \rightarrow 0} \frac{\sin(8x)}{\sin(4x)}
\][/tex]
4. Introduce a Common Factor:
Rewrite the numerator and the denominator in a form that separates out [tex]\(x\)[/tex], allowing us to use the basic trigonometric limit:
[tex]\[
\lim_{x \rightarrow 0} \frac{\sin(8x)}{\sin(4x)} = \lim_{x \rightarrow 0} \frac{\frac{\sin(8x)}{8x} \cdot 8x}{\frac{\sin(4x)}{4x} \cdot 4x}
\][/tex]
5. Apply the Fundamental Trigonometric Limit:
We know from the trigonometric limit property:
[tex]\[
\lim_{x \rightarrow 0} \frac{\sin(8x)}{8x} = 1 \quad \text{and} \quad \lim_{x \rightarrow 0} \frac{\sin(4x)}{4x} = 1
\][/tex]
Substitute these limits into the expression:
[tex]\[
\lim_{x \rightarrow 0} \frac{\left(1 \cdot 8x\right)}{\left(1 \cdot 4x\right)}
\][/tex]
6. Simplify:
Simplify the fraction we are left with:
[tex]\[
\lim_{x \rightarrow 0} \frac{8x}{4x} = \lim_{x \rightarrow 0} 2 = 2
\][/tex]
Hence, the solution to the limit is:
[tex]\[
\lim _{x \rightarrow 0} \frac{\sin(8x)}{\sin(4x)} = 2
\][/tex]
