Solving a Limit Problem: $\lim_{x \to 0} \frac{x\cos(x)}{\sin(x)}$

[Petrus]

Hello,
I got problem with understanding one exemple
$\lim_{x \to 0} \frac{x\cos(x)}{\sin(x)}$ = $\lim_{x \to 0}\frac{\cos(x)}{\sin(x)}$
if i do it backway i can see that correct with it $\frac{a/b}{c/d}$is equal to $\frac{ad}{bc}$ then i start to do the way what i type and don't get correct. Can anyone possible try explain for me thanks.(Sorry about bad title I don't know what I should name it)

 

[MarkFL]

Re: equal,limit,derivate

Hello Petrus,

You post is simply unreadable to me. Can you edit it, using the backslash "\" before the frac commands (and trig functions) to make it understandable?

 

[I like Serena]

Re: equal,limit,derivate

Hello,
I got problem with understanding one exemple
lim x->0 $/frac{xcos(x)}{sin(x)} = lim x->0 $/frac{cos(x)}{sinx/x)}
if i do it backway i can see that correct with $/frace{a/b}{b/c}$=${ad}{bc}$ then i start to do the way what i type and don't get correct. Can anyone possible try explain for me thanks.(Sorry about bad title I don't know what I should name it)

Hi Petrus!

Well... it is painfully obvious that the latex expressions are not working for you. ;-)
So I'll try to do without.

When you say (a/b) / (b/c) = (ad) / (bc) that is not correct.
It should be: (a/b) / (b/c) = (a/b) * (c/b) = (ac) / (b^2).

Note that dividing by a fraction is the same as multiplying by its inverse.
And also that multiplying 2 fractions means to multiply the numerators and separately the denominators.

To get back to your original expression, you have:

cos(x) / (sin(x) / x) = cos(x) * (x / sin(x)) = (cos(x) * x) / sin(x) = (x cos(x)) / sin(x).

 

[Petrus]

Re: equal,limit,derivate

Hi Petrus!

Well... it is painfully obvious that the latex expressions are not working for you. ;-)
So I'll try to do without.

When you say (a/b) / (b/c) = (ad) / (bc) that is not correct.
It should be: (a/b) / (b/c) = (a/b) * (c/b) = (ac) / (b^2).

Note that dividing by a fraction is the same as multiplying by its inverse.
And also that multiplying 2 fractions means to multiply the numerators and separately the denominators.

To get back to your original expression, you have:

cos(x) / (sin(x) / x) = cos(x) * (x / sin(x)) = (cos(x) * x) / sin(x) = (x cos(x)) / sin(x).

Now it make Clear! Thanks!

 

[I like Serena]

Re: equal,limit,derivate

Now it make Clear! Thanks!

Good! ;)

For later reference: you can use \lim_{x \to 0} to format your limit nicely:
$$\lim_{x \to 0}$$