To determine the limit
[tex]\[ \lim_{x \rightarrow 0} \frac{\sin(5x)}{x}, \][/tex]
we can proceed by using some fundamental limit properties. Let's analyze the expression step-by-step.
First, we'll recall the well-known limit result:
[tex]\[ \lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1. \][/tex]
Given this information, we need to modify our limit to fit this form. Let's rewrite [tex]\(\frac{\sin(5x)}{x}\)[/tex] in a more manageable way.
Notice that:
[tex]\[ \frac{\sin(5x)}{x} = \frac{\sin(5x)}{5x} \cdot 5. \][/tex]
Now, we can separate the constant factor and the limit:
[tex]\[ \frac{\sin(5x)}{x} = 5 \left( \frac{\sin(5x)}{5x} \right). \][/tex]
Observe that as [tex]\( x \)[/tex] approaches [tex]\( 0 \)[/tex], [tex]\( 5x \)[/tex] also approaches [tex]\( 0 \)[/tex]. Therefore, using the fact that:
[tex]\[ \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \][/tex]
we can substitute [tex]\( 5x \)[/tex] for [tex]\( u \)[/tex]:
[tex]\[ \lim_{x \to 0} \frac{\sin(5x)}{5x} = \lim_{u \to 0} \frac{\sin(u)}{u} = 1. \][/tex]
Thus, our limit expression becomes:
[tex]\[ 5 \left( \lim_{x \to 0} \frac{\sin(5x)}{5x} \right) = 5 \cdot 1 = 5. \][/tex]
Therefore:
[tex]\[ \lim_{x \rightarrow 0} \frac{\sin(5x)}{x} = 5. \][/tex]
So, the correct answer is [tex]\( 5 \)[/tex].
