Questions: Evaluate the limit. lim as x approaches 0 of (sin 8x)/(sin 7x) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. lim as x approaches 0 of (sin 8x)/(sin 7x) = 8/7 B. The limit is undefined.

Evaluate the limit.
lim as x approaches 0 of (sin 8x)/(sin 7x)

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. lim as x approaches 0 of (sin 8x)/(sin 7x) = 8/7
B. The limit is undefined.

Transcript text: Evaluate the limit. \[ \lim _{x \rightarrow 0} \frac{\sin 8 x}{\sin 7 x} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\lim _{x \rightarrow 0} \frac{\sin 8 x}{\sin 7 x}=\frac{8}{7}$ B. The limit is undefined.

Solution

Solution Steps

To evaluate the limit $\lim _{x \rightarrow 0} \frac{\sin 8x}{\sin 7x}$, we can use the standard limit property $\lim_{x \to 0} \frac{\sin kx}{kx} = 1$. By rewriting the expression in terms of this property, we can simplify the limit to a form that is easy to evaluate.

Solution Approach

Rewrite the expression using the property $\lim_{x \to 0} \frac{\sin kx}{kx} = 1$.
Simplify the expression to find the limit.

Step 1: Apply the Limit Property

To evaluate the limit $\lim _{x \rightarrow 0} \frac{\sin 8x}{\sin 7x}$, we use the standard limit property $\lim_{x \to 0} \frac{\sin kx}{kx} = 1$. This property allows us to rewrite the expression in a form that is easier to evaluate.

Step 2: Rewrite the Expression

We can express the limit as: \[ \lim_{x \to 0} \frac{\sin 8x}{\sin 7x} = \lim_{x \to 0} \frac{\frac{\sin 8x}{8x}}{\frac{\sin 7x}{7x}} \cdot \frac{8x}{7x} \]

Step 3: Simplify the Expression

Using the property $\lim_{x \to 0} \frac{\sin kx}{kx} = 1$, we simplify the expression: \[ = \frac{8}{7} \cdot \frac{1}{1} = \frac{8}{7} \]