Limit Comprehension Question.

The limit problem is as follows: $\displaystyle \lim_{x\to\0}\frac{11x}{tan21x}$.

Some thoughts I had about solving this were to express the inverse of tangent as a fraction of cosine divided by sine: $\displaystyle \frac{cos11x}{sin21x}$. By doing this I thought that I could then use the $\displaystyle \lim_{x\to\0}\frac{sin(x)}{x}=1$ theorem(?) to multiply the numerator by 21 yielding $\displaystyle \frac{21x}{sin21x}$ which equals one. But I don't really know what to do with cosine and this is where I am stuck. The answer in the book is $\displaystyle \frac{11}{21}$. If someone could help point me in the right direction I would really appreciate it.

(P.S., sorry if this is in the wrong forum)

Re: Limit Comprehension Question.

You solved it already.

since $\displaystyle tan(21x) = \frac{\sin(21x)}{\cos(21x)} $ you end up with

$\displaystyle \lim_{x\to\0}{\frac{11x*cos21x}{sin21x}}$

$\displaystyle = 11* \lim_{x\to\0}{\frac{ 21x *\cos(21x)}{21*\sin(21x)}}$

$\displaystyle = 11* \lim_{x\to\0}{\frac{1* \cos(21x)}{21}} $ .. .. .. .. Since you said $\displaystyle \lim_{x\to\0} {\frac{21x}{sin(21x} = 1$

$\displaystyle = 11*\frac{1}{21} * \lim_{x\to\0}{\cos(21x)}$ .. .. .. .. since $\displaystyle \cos(21*0) = 1$

$\displaystyle =\frac{11}{21}$

Have a good day

Re: Limit Comprehension Question.

Hello, KhanDisciple!

You have the right idea.

We just need some Olympic-level gymnastics.

Quote:

$\displaystyle \lim_{x\to0}\frac{11x}{\tan(21x)}$

We have: .$\displaystyle \frac{11x}{\tan(21x)} \;=\; \frac{11x}{\frac{\sin(21x)}{\cos(21x)}} \;=\;\frac{11x\cos(21x)}{\sin(21x)} $

Multiply by $\displaystyle \frac{21}{21}:\;\;\frac{21}{21}\cdot\frac{11x\cos( 21x)}{\sin(21x)} \;=\;\frac{11}{21}\cdot\frac{21x}{\sin(21x)}\cdot \cos(21x) $

Therefore: .$\displaystyle \lim_{x\to0}\left[\frac{11}{21}\cdot\frac{21x}{\sin(21x)}\cdot\cos(2 1x)\right] \;=\;\frac{11}{21}\cdot 1 \cdot 1 \;=\;\frac{11}{21} $

Re: Limit Comprehension Question.

Quote:

Originally Posted by

**KhanDisciple** The limit problem is as follows: $\displaystyle \lim_{x\to\0}\frac{11x}{tan21x}$.

Some thoughts I had about solving this were to express the inverse of tangent as a fraction of cosine divided by sine: $\displaystyle \frac{cos11x}{sin21x}$. By doing this I thought that I could then use the $\displaystyle \lim_{x\to\0}\frac{sin(x)}{x}=1$ theorem(?) to multiply the numerator by 21 yielding $\displaystyle \frac{21x}{sin21x}$ which equals one. But I don't really know what to do with cosine and this is where I am stuck. The answer in the book is $\displaystyle \frac{11}{21}$. If someone could help point me in the right direction I would really appreciate it.

(P.S., sorry if this is in the wrong forum)

If you didn't want to perform the Olympic Level gymnastics, L'Hospital's Rule simplifies things greatly since this goes to $\displaystyle \displaystyle \begin{align*} \frac{0}{0} \end{align*}$.

$\displaystyle \displaystyle \begin{align*} \lim_{x \to 0}\frac{11x}{\tan{\left( 21x \right)}} &= \lim_{x \to 0}\frac{\frac{d}{dx} \left( 11x \right)}{\frac{d}{dx} \left[ \tan{\left( 21x \right)} \right]} \\ &= \lim_{x \to 0} \frac{11}{21\sec^2{\left( 21x \right)}} \\ &= \frac{11}{21} \end{align*}$

Re: Limit Comprehension Question.

Thanks a lot guys, you are the best!