Question

find the limit using $lim_{\theta
ightarrow0}\frac{sin\theta}{\theta}=1$.
$lim_{x
ightarrow0}\frac{xcsc10x}{cos17x}$
select the correct choice below and, if necessary, fill in the answer box in your choice.
a. $lim_{x
ightarrow0}\frac{xcsc10x}{cos17x}=$ (simplify your answer.)
b. the limit does not exist.

Explanation:

Step1: Rewrite csc function

Recall that $\csc(10x)=\frac{1}{\sin(10x)}$. So the limit $\lim_{x
ightarrow0}\frac{x\csc(10x)}{\cos(17x)}$ becomes $\lim_{x
ightarrow0}\frac{x}{\sin(10x)\cos(17x)}$.

Step2: Manipulate the expression

Multiply and divide by 10: $\lim_{x
ightarrow0}\frac{1}{10}\cdot\frac{10x}{\sin(10x)}\cdot\frac{1}{\cos(17x)}$.

Step3: Use the limit property $\lim_{\theta

ightarrow0}\frac{\sin\theta}{\theta} = 1$
Let $\theta = 10x$. As $x
ightarrow0$, $\theta
ightarrow0$. Also, $\lim_{x
ightarrow0}\cos(17x)=\cos(0) = 1$. We know that $\lim_{\theta
ightarrow0}\frac{\sin\theta}{\theta}=1$, so $\lim_{x
ightarrow0}\frac{10x}{\sin(10x)} = 1$. Then $\lim_{x
ightarrow0}\frac{1}{10}\cdot\frac{10x}{\sin(10x)}\cdot\frac{1}{\cos(17x)}=\frac{1}{10}\times1\times\frac{1}{1}$.

Answer:

A. $\lim_{x
ightarrow0}\frac{x\csc(10x)}{\cos(17x)}=\frac{1}{10}$