To find the limit of [tex]\(\lim_{x \rightarrow 0} \frac{\sin 7x}{3x}\)[/tex], we'll use well-established properties of limits and the behavior of the sine function near zero.
1. Understand the Limit Setup:
The expression we want to find the limit of as [tex]\(x\)[/tex] approaches 0 is [tex]\(\frac{\sin 7x}{3x}\)[/tex].
2. Key Limit Property:
Recall the fundamental trigonometric limit:
[tex]\[
\lim_{t \rightarrow 0} \frac{\sin t}{t} = 1
\][/tex]
We can use this property by manipulating our given limit to match this form.
3. Rewrite the Limit:
In order to use the fundamental limit property, we can rewrite [tex]\(\frac{\sin 7x}{3x}\)[/tex] in a more suitable form.
[tex]\[
\lim_{x \rightarrow 0} \frac{\sin 7x}{3x}
\][/tex]
Notice that [tex]\( \sin 7x \)[/tex] involves a sine function argument of [tex]\(7x\)[/tex]. We want to separate this in a way that lets us use the [tex]\(\frac{\sin t}{t}\)[/tex] property.
4. Factor and Simplify:
Break down the expression:
[tex]\[
\frac{\sin 7x}{3x} = \frac{7 \sin 7x}{7 \cdot 3x} = \frac{7}{3} \cdot \frac{\sin 7x}{7x}
\][/tex]
5. Apply the Key Limit Property:
Now we can use the limit property on [tex]\(\frac{\sin 7x}{7x}\)[/tex]. As [tex]\(x\)[/tex] approaches 0, [tex]\(7x\)[/tex] also approaches 0. Therefore:
[tex]\[
\lim_{x \rightarrow 0} \frac{\sin 7x}{7x} = 1
\][/tex]
6. Combine the Results:
Using the given property,
[tex]\[
\lim_{x \rightarrow 0} \left( \frac{7}{3} \cdot \frac{\sin 7x}{7x} \right) = \frac{7}{3} \cdot 1 = \frac{7}{3}
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{x \rightarrow 0} \frac{\sin 7x}{3x} = \frac{7}{3}
\][/tex]
