Finding this limit involving sin and cos

In summary, the conversation discusses a function and its limit, and the use of L'Hopital's Rule to find the limit. The correct application of L'Hopital's Rule is discussed as well as an alternative method using Taylor series.
  • #1
lo2
55
0

Homework Statement



I have this function:

[itex]f(x) = \frac{1}{x}-\frac{\cos{(x)}}{\sin{(x)}}[/itex]

For all [itex]x \in R[/itex] where [itex] x \neq n \pi, n \in Z [/itex]

Ok I have to find the following limit:

[itex]lim_{x\rightarrow0+}(f(x))[/itex]

Homework Equations



Limits in general and perhaps the always great Hospital's rule.


The Attempt at a Solution



I have tried to put on the same fraction line:

[itex]f(x) = \frac{\sin{(x)}-x\cos{(x)}}{x\sin{(x)}}[/itex]

And then using the Hospital rule, but it does not really seem to bring me any further...

The first derivative of it is:

[itex]f(x) = \frac{x^2-1+(\cos{(x)})^2}{x^2(\sin{(x)})^2}[/itex]

And then I could use the Hospital rule again but it just seems as though it will make it worse, the sinus will always be in the denominator.
 
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  • #2
lo2 said:

Homework Statement



I have this function:

[itex]f(x) = \frac{1}{x}-\frac{\cos{(x)}}{\sin{(x)}}[/itex]

For all [itex]x \in R[/itex] where [itex] x \neq n \pi, n \in Z [/itex]

Ok I have to find the following limit:

[itex]lim_{x\rightarrow0+}(f(x))[/itex]

Homework Equations



Limits in general and perhaps the always great Hospital's rule.

The Attempt at a Solution



I have tried to put on the same fraction line:

[itex]f(x) = \frac{\sin{(x)}-x\cos{(x)}}{x\sin{(x)}}[/itex]

And then using the Hospital rule, but it does not really seem to bring me any further...

The first derivative of it is:

[itex]f(x) = \frac{x^2-1+(\cos{(x)})^2}{x^2(\sin{(x)})^2}[/itex]

And then I could use the Hospital rule again but it just seems as though it will make it worse, the sinus will always be in the denominator.

First of all, you're not applying L'Hopital's Rule correctly. You're supposed to differentiate the numerator and denominator *separately*. Instead you differentiated the whole expression using the quotient rule. (BTW, even that expression you got can be further simplified. Use [itex]\sin^2 x + \cos^2 x = 1[/itex] on the numerator. Irrelevant to the question, but something you should take note of).

After you apply LHR correctly, take the reciprocal of the expression and see what it reduces to.

Alternatively, you can just apply the Taylor series throughout and get the answer quickly without using LHR.
 
  • #3
Ah yeah ok, I can see that I have used LHR wrongly...

I think I have got the right answer applying the rule correctly! So thanks a lot :)
 
  • #4
lo2 said:
Ah yeah ok, I can see that I have used LHR wrongly...

I think I have got the right answer applying the rule correctly! So thanks a lot :)

You're welcome. :smile:
 

1. What is the definition of a limit involving sin and cos?

The limit involving sin and cos is the value that a function approaches as the input value approaches a specific value. It is denoted as lim x→a f(x) and can be found by evaluating the function at values increasingly closer to the specific value.

2. How do I find the limit of a function involving sin and cos?

To find the limit of a function involving sin and cos, you can use algebraic manipulation, trigonometric identities, and the properties of limits. You can also use a graphing calculator or an online calculator to find the limit numerically.

3. What are the common mistakes to avoid when finding limits involving sin and cos?

Some common mistakes to avoid when finding limits involving sin and cos include forgetting to simplify the expression before taking the limit, using the wrong trigonometric identity, and not considering the properties of limits such as the limit of a sum being the sum of the limits.

4. Can limits involving sin and cos have multiple solutions?

Yes, limits involving sin and cos can have multiple solutions. This is because trigonometric functions are periodic and have repeating patterns. It is important to consider the domain and range of the function to determine the appropriate solution for the limit.

5. How can limits involving sin and cos be applied in real life?

Limits involving sin and cos can be used in various fields such as physics, engineering, and economics. For example, they can be used to model the motion of a pendulum, the vibrations of a guitar string, or the fluctuations in stock prices. They can also be used to determine the maximum and minimum values of a function, which is important in optimization problems.

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