Question

exercises 2.3 the limit laws score: 15.75/21 answered: 16/21 question textbook videos + evaluate: $lim_{\theta
ightarrow0}\frac{sin(13\theta)}{sin(10\theta)}$ question help: message instructor submit question jump to answer

Explanation:

Step1: Use the limit - rule $\lim_{x

ightarrow0}\frac{\sin(ax)}{ax}=1$
Rewrite $\lim_{\theta
ightarrow0}\frac{\sin(13\theta)}{\sin(10\theta)}$ as $\lim_{\theta
ightarrow0}\frac{\sin(13\theta)}{13\theta}\cdot\frac{10\theta}{\sin(10\theta)}\cdot\frac{13\theta}{10\theta}$.

Step2: Apply the limit - rule

We know that $\lim_{\theta
ightarrow0}\frac{\sin(13\theta)}{13\theta}=1$ and $\lim_{\theta
ightarrow0}\frac{\sin(10\theta)}{10\theta}=1$. So, $\lim_{\theta
ightarrow0}\frac{\sin(13\theta)}{13\theta}\cdot\frac{10\theta}{\sin(10\theta)}\cdot\frac{13\theta}{10\theta}=\lim_{\theta
ightarrow0}\frac{\sin(13\theta)}{13\theta}\cdot\frac{1}{\lim_{\theta
ightarrow0}\frac{\sin(10\theta)}{10\theta}}\cdot\frac{13}{10}$.

Step3: Calculate the limit

Since $\lim_{\theta
ightarrow0}\frac{\sin(13\theta)}{13\theta}=1$ and $\lim_{\theta
ightarrow0}\frac{\sin(10\theta)}{10\theta}=1$, then $\lim_{\theta
ightarrow0}\frac{\sin(13\theta)}{\sin(10\theta)}=\frac{13}{10}$.

Answer:

$\frac{13}{10}$