Question

What did I do wrong in this problem?
(Finding limits with trigonometric functions)

Answer 1

I know my finaly answer is wrong, but I can't figure out what I did wrong. I know there are other ways to solve it, but what exactly is invalid about the steps i took.

Answer 2

HI!

Answer 3

hey :D

Answer 4

before we begin, is it clear that the answer is \(\frac{2}{4}=\frac{1}{2}\) ? they are after all the only numbers you see

Answer 5

Yeah, I'm pretty sure looking at the graph the final answer should be 0.5. However the way I did it, my answer comes to 0.

Answer 6

this is what you know:\[\lim_{x\to 0}\frac{\sin(x)}{x}=1\]

Answer 7

and therefore
\[\lim_{x\to 0}\frac{\sin(ax)}{ax}=1\] as well since they are the same thing

Answer 8

so the gimmick here is to write
\[\lim_{x\to 0}\frac{\sin(ax)}{\sin(bx)}=\lim_{x\to 0}\frac{\sin(ax)}{ax}\frac{bx}{\sin(bx)}\times \frac{ax}{bx}\]

Answer 9

you get \[1\times 1\times \frac{a}{b}\]

Answer 10

which in your specific example is \(\frac{2}{4}\)

Answer 11

Oh I see. I think I factored out the limits in the wrong way. Alright thanks for the answer

Answer 12

\[\color\magenta\heartsuit\]