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

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In summary, Petrus is having difficulty understanding one example. He is trying to use latex expressions, but they are not working. He is then told to do without, and is provided with a reference to how to format a limit nicely.
  • #1
Petrus
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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)
 
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  • #2
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?
 
  • #3
Re: equal,limit,derivate

Petrus said:
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).
 
  • #4
Re: equal,limit,derivate

I like Serena said:
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!
 
  • #5
Re: equal,limit,derivate

Petrus said:
Now it make Clear! Thanks!

Good! ;)

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

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

What is a limit problem?

A limit problem is a type of mathematical problem in which the value of a function or expression is approached as the input variable approaches a specific value or "limit".

Why is solving limit problems important?

Solving limit problems is important in many areas of mathematics and science because it allows us to understand the behavior of functions and expressions as they approach specific values. This information can be used to make predictions and solve real-world problems.

What does the notation $\lim_{x \to 0}$ mean?

The notation $\lim_{x \to 0}$ represents the limit of a function or expression as the input variable, in this case x, approaches a specific value, in this case 0. It is read as "the limit of x as it approaches 0".

How do you solve a limit problem?

To solve a limit problem, you need to first determine the type of problem it is (e.g. indeterminate form, one-sided limit, etc.) and then apply the appropriate techniques, such as algebraic manipulation, factoring, or substitution, to simplify the expression and evaluate the limit.

What is the solution to the limit problem $\lim_{x \to 0} \frac{x\cos(x)}{\sin(x)}$?

The solution to this limit problem is 1. By applying the algebraic identity $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$, the expression can be simplified to $\lim_{x \to 0} \frac{\cos(x)}{1} = \cos(0) = 1$. Therefore, the limit as x approaches 0 of the given expression is equal to 1.

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