Math Snap

STEP 1

1. The limit involves trigonometric functions and requires simplification.
2. The expression $\frac{\sin^2(8x)}{1-\cos(8x)}$ can be simplified using trigonometric identities.
3. Multiplying by a conjugate can help simplify the expression.

STEP 2

1. Identify the conjugate to multiply the expression.
2. Simplify the expression using trigonometric identities.
3. Evaluate the limit.

STEP 3

To simplify the expression, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $1 - \cos(8x)$ is $1 + \cos(8x)$. Therefore, multiply by:
$$\frac{1 + \cos(8x)}{1 + \cos(8x)}$$

STEP 4

Simplify the expression after multiplying by the conjugate:
$$\frac{\sin^2(8x)(1 + \cos(8x))}{(1 - \cos(8x))(1 + \cos(8x))}$$ The denominator becomes:
$$(1 - \cos(8x))(1 + \cos(8x)) = 1 - \cos^2(8x) = \sin^2(8x)$$ So the expression simplifies to:
$$\frac{\sin^2(8x)(1 + \cos(8x))}{\sin^2(8x)}$$ Cancel $\sin^2(8x)$ from the numerator and denominator:
$$1 + \cos(8x)$$

Now evaluate the limit as $x \rightarrow 0$:
$$\lim_{x \rightarrow 0} (1 + \cos(8x))$$ Since $\cos(8x)$ approaches $\cos(0) = 1$ as $x \rightarrow 0$, the limit is:
$$1 + 1 = 2$$ The expression to multiply by is:
$$\boxed{1 + \cos(8x)}$$ The value of the limit is:
$$\boxed{2}$$