Question

limit x proches to zero sin5x/sin4x

Answer 1

Hey there :) Welcome to OpenStudy!

Answer 2

\[\Large\rm \lim_{x\to0}\frac{\sin(5x)}{\sin(4x)}\]Mmm this one is kinda fun.

Answer 3

If you split it up like this, and deal with each separtely,\[\Large\rm =\lim_{x\to0}\frac{\sin(5x)}{1}\cdot \lim_{x\to0}\frac{1}{\sin(4x)}\]it might be more manageable.

Answer 4

Recall this limit:\[\Large\rm \lim_{x\to0}\frac{\sin(\color{orangered}{x})}{\color{orangered}{x}}=1\]When the angle on top matches the value in the bottom, we get this nice identity.

Answer 5

So for our problem, we'll multiply the top and bottom by x.
We'll give one to the first limit, and the other to the other limit.\[\Large\rm =\frac{x}{x}\left(\lim_{x\to0}\frac{\sin(5x)}{1}\cdot \lim_{x\to0}\frac{1}{\sin(4x)}\right)\]
\[\Large\rm =\lim_{x\to0}\frac{\sin(5x)}{x}\cdot \lim_{x\to0}\frac{x}{\sin(4x)}\]

Answer 6

Mmm looking at the first limit, our value in the bottom still doesn't match :o
let's multiply the top and bottom by 5,\[\Large\rm =\frac{5}{5}\cdot\lim_{x\to0}\frac{\sin(5x)}{x}\cdot \lim_{x\to0}\frac{x}{\sin(4x)}\]

Answer 7

\[\Large\rm =5\cdot\lim_{x\to0}\frac{\sin(\color{orangered}{5x})}{\color{orangered}{5x}}\cdot \lim_{x\to0}\frac{x}{\sin(4x)}\]

Answer 8

Ah yes! We've done it!
Now we can apply our limit identity.
That first limit is approaching 1.

Understand how to do similar with the other limit?