Question

using lim (radian ->0) (sin (radian))/(radian) = 1, find the limit of lim (t->0) (2t)/(tan t).

Answer 1

split tan into sin and cos and it's pretty straightforward

Answer 2

also you shouldn't write sin(radian), that doesn't make a lot of sense...

Answer 3

sine and cosine are functions of numbers, and as such they correspond to the right triangle trigonometry if you measure an angle in "radians"
but if you write \(f(x)=x^2\) no one is going to ask you if \(x\) is in degrees, radians, Celsius, Fahrenheit or anything else. it is presumed to be a real number

Answer 4

\[\frac{2t}{\tan(t)}=\frac{2t\cos(t)}{\sin(t)}=2\times \frac{t}{\sin(t)}\times \cos(t)\] now take the limit piece by piece

Answer 5

sorry, that's how it was in the book, they called theta radians, but there's no place to add the symbol raidan when asking the original question, as you may have noticed.
using \[\lim_{\theta \rightarrow 0} \frac{ \sin \theta }{ \theta }\]
solve for
\[\lim_{t \rightarrow 0} \frac{ 2t }{ \tan t }\]
so I get
\[\lim_{t \rightarrow 0} \frac{ 2t }{ \frac{ \sin t }{ \cos t } }\]
Sorry, but it's not actually straight forward for me TuringTest, it's why I'm asking ^_^

Answer 6

satellite showed ou what I was trying to suggest to you\[\frac{2t}{\tan(t)}=\frac{2t\cos(t)}{\sin(t)}=2\times \frac{t}{\sin(t)}\times \cos(t)\]

Answer 7

you*

Answer 8

in case you have some doubt as to the limit of the middle term perhaps you may prefer it as\[\frac{2t}{\tan(t)}=\frac{2t\cos(t)}{\sin(t)}=2\times {\frac1{\sin t\over t}}\times \cos(t)\]

Answer 9

i would avoid writing compuund fractions if at all possible because they are confusing. instead of writing
\[\frac{ 2t }{ \frac{ \sin t }{ \cos t } }\] it is best to go right to
\[\frac{2t\cos(t)}{\sin(t)}\]

Answer 10

So you know how to evaluate the following limits?
\[\lim_{t \rightarrow 0}\frac{t}{\sin(t)}=? ; \lim_{t \rightarrow 0}\cos(t)=?\]
Are there any questions with what sat and turing said?

Answer 11

I believe we have lost our asker...

Answer 12

I believe the asker had to go to class.
And I prefer writing out every single step, including compound fractions. It makes it clearer to me wtf went down in each step.