Find the following limit using $$ \lim _{ heta ightarrow 0} \frac{\sin heta}{ heta}=1 $$. $$ \lim _{x ightarrow 0} \frac{ an 5 x}{\sin 9 x} $$ Select the correct choice below and, if necessary, fill in the answer box in your choice. A. $$ \lim _{x ightarrow 0} \frac{ an 5 x}{\sin 9 x}=\square $$ (Type an integer or a simplified fraction.) B. The limit does not exist.

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Find the following limit using $$ \lim _{	heta ightarrow 0} \frac{\sin 	heta}{	heta}=1 $$. $$ \lim _{x ightarrow 0} \frac{	an 5 x}{\sin 9 x} $$ Select the correct choice below and, if necessary, fill in the answer box in your choice. A. $$ \lim _{x ightarrow 0} \frac{	an 5 x}{\sin 9 x}=\square $$ (Type an integer or a simplified fraction.) B. The limit does not exist.

UpStudy AI · Accepted Answer

To find the limit:
$$
\lim _{x \rightarrow 0} \frac{\tan 5x}{\sin 9x}
$$
we can use the known limit $$\lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1$$ and the fact that $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$.

Start by rewriting the expression:
$$
\frac{\tan 5x}{\sin 9x} = \frac{\sin 5x}{\cos 5x} \times \frac{1}{\sin 9x}
$$

Now, express the limit in a form that leverages the standard limit:
$$
\lim_{x \rightarrow 0} \frac{\tan 5x}{\sin 9x} = \lim_{x \rightarrow 0} \left( \frac{\sin 5x}{5x} \times \frac{5x}{\sin 9x} \times \frac{1}{\cos 5x} \right)
$$

Evaluating each part separately:
1. $$\lim_{x \rightarrow 0} \frac{\sin 5x}{5x} = 1$$
2. $$\lim_{x \rightarrow 0} \frac{5x}{\sin 9x} = \frac{5}{9}$$ (since $$\lim_{x \rightarrow 0} \frac{\sin 9x}{9x} = 1$$)
3. $$\lim_{x \rightarrow 0} \frac{1}{\cos 5x} = 1$$ (because $$\cos 0 = 1$$)

Multiplying these results together:
$$
1 \times \frac{5}{9} \times 1 = \frac{5}{9}
$$

**Answer:**

**A.** $$ \lim _{x \rightarrow 0} \frac{\tan 5 x}{\sin 9 x}= \frac{5}{9} $$
The limit is $$ \frac{5}{9} $$.

Find the following limit using .

Select the correct choice below and, if necessary, fill in the answer box in your choice.
A. (Type an integer or a simplified fraction.)
B. The limit does not exist.

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Find the following limit using . Select the correct choice below and, if necessary, fill in the answer box in your choice. A. (Type an integer or a simplified fraction.) B. The limit does not exist.