Question

Medal will be awarded (Calculus)

Answer 1

\[\lim_{x \rightarrow 0}\frac{ 1 }{ x \csc x }\]

Answer 2

Compute the limit

Answer 3

Hey Cimber :)
Hmm that name sounds familiar...
wasn't that the name of that adorable lil cub from the Lion King?

Answer 4

Let's rewrite cosecant in terms of sine, that should help out quite a bit.

Answer 5

That's Simba but thanks :)

Answer 6

Ooo ya >.<

Answer 7

\[\Large\rm \csc x=\frac{1}{\sin x}\]Remember this one? :d

Answer 8

Yes, but would I flip it then because it would turn to be \[1/1/x sinx?\]

Answer 9

then it would just be x sinx

Answer 10

Careful! :)
Only the sin x is getting flipped, ya?\[\Large\rm \lim_{x \rightarrow 0}\frac{ 1 }{ x \csc x }=\quad \lim_{x\to0}\frac{1}{x\left(\frac{1}{\sin x}\right)}\]

Answer 11

You can think of it like this maybe: 1/x * 1/(1/sinx)
or like this: 1/(x/sinx)

Answer 12

Why though

Answer 13

\[\Large\rm \frac{1}{x\color{orangered}{\csc x}}=\frac{1}{x\color{orangered}{\left(\frac{1}{\sin x}\right)}}=\frac{1}{\color{orangered}{\left(\frac{x}{\sin x}\right)}}\]The x doesn't go in the bottom bottom with the sinx.
Is that the step we're having trouble with?

Answer 14

oooooooooooh

Answer 15

No I'm just having trouble all over because my final exam is next week and it's cumulative

Answer 16

So what is the final answer then??

Answer 17

\[\Large\rm \frac{1}{x\color{orangered}{\csc x}}=\frac{1}{x\color{orangered}{\left(\frac{1}{\sin x}\right)}}=\frac{1}{\color{orangered}{\left(\frac{x}{\sin x}\right)}}=\frac{\sin x}{x}\]So it'll flip like that, ya? :)

Answer 18

And uhh, this is just a limit identity that you're suppose to remember :P\[\Large\rm \lim_{x\to0}\frac{\sin x}{x}=1\]

Answer 19

Ohh I see now. Thank you :)