Questions: Evaluate the limit lim as x approaches 0 of (sin 4x)/(9x)

Solution Steps

To evaluate the limit \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{9 x}\), we can use the standard limit property \(\lim_{x \to 0} \frac{\sin x}{x} = 1\). By manipulating the given expression to match this form, we can find the limit.

We start with the limit expression: \[ \lim_{x \rightarrow 0} \frac{\sin(4x)}{9x} \]

Using the known limit property \(\lim_{x \to 0} \frac{\sin(kx)}{kx} = 1\), we can rewrite our limit: \[ \lim_{x \rightarrow 0} \frac{\sin(4x)}{9x} = \lim_{x \rightarrow 0} \frac{\sin(4x)}{4x} \cdot \frac{4}{9} \] As \(x\) approaches \(0\), \(\frac{\sin(4x)}{4x} \rightarrow 1\).

Thus, we have: \[ \lim_{x \rightarrow 0} \frac{\sin(4x)}{9x} = 1 \cdot \frac{4}{9} = \frac{4}{9} \]

Final Answer