Q. 24 - Calculus

Question

Find the limit by hand.

$\underset{x\to 0}{\lim }\frac{{\sin }^2 3x}{x}$

Step-by-Step Solution

Answer

On solving the limit, we get, $0$.

Step 1. Given information.

Consider the given question,

$\underset{x\to 0}{\lim }\frac{{\sin }^2 3x}{x}$

Step 2. Solve the given limit.

Rewriting the expression,

$$\begin{gathered} =\underset{x\to 0}{\lim }\left(\frac{\sin 3x \sin 3x}{3x}·3\right) \\ =3\underset{x\to 0}{\lim }\left(\frac{\sin 3x}{3x}\right)\underset{x\to 0}{\lim }\sin 3x \\ =3\left(1\right) \sin \left(3·0\right) \\ =3 \sin 0 \\ =3\left(0\right) \\ =0 \end{gathered}$$