Finding the limit of a trig function

In summary, the conversation is about finding the limit of the function \lim_{x\rightarrow 0^{+}}(\frac{\csc2x}{x}) and various methods are suggested, such as using l'Hopital's rule or the squeeze theorem. However, it is pointed out that the limit cannot be solved using those methods since it does not have the form 0/0 or inf/inf. The person also suggests using the identity \lim_{x\rightarrow c} \sin x = \sin c to solve the limit, but ultimately concludes that the limit is undefined as it results in 1/0.
  • #1
lLovePhysics
169
0
I'm stuck on this limit function and I don't know what to do next. Please help me out. Thanks!

[tex]\lim_{x\rightarrow 0^{+}}(\frac{\csc2x}{x})[/tex]

I just turned csc2x into 1\sin2x so then I have:

[tex] \lim_{x\rightarrow 0^{+}} (\frac{1}{x\sin2x})[/tex]

Then I used the trig identity: sin2x=2sinxcosx

...but I'm not sure if this is going to take me anywhere.

I know that [tex]\lim_{x\rightarrow c} \sin x = \sin c[/tex] but I'm not sure how to incoporate that identity in this problem.
 
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  • #2
You should use l'Hopital's rule. Or just note that 1/0 is never going to be a good thing...
 
  • #3
Would the squeeze theorem work? -1<sinxcosx<1 to prove that the limit = 0?
 
  • #4
The easiest way is genneths second comment. Just realize that the denominator, x sin 2x, is going to 0, and there is nothing on the numerator to cancel out with. Denominator approaches 0, whole thing approach infinity.
 
  • #5
You can t use L'Hospitals rule. It has to be of the form 0/0 or inf/inf. Since this is neither, try to use the identity lim x->0 sinx/x =1.
 
  • #6
To be honest, this is one of those limits which doesn't need a limit. You just put zero in, and go, "oh, it's 1/0". No funky limit taking will change that.
 

What is a limit of a trigonometric function?

A limit of a trigonometric function is a value that a function approaches as the input approaches a certain value. It represents the behavior of the function near that specific value.

How do you find the limit of a trigonometric function?

To find the limit of a trigonometric function, you can use algebraic manipulation, graphing, or substitution. You can also use trigonometric identities and properties to simplify the function and evaluate the limit.

Why is it important to find the limit of a trigonometric function?

Finding the limit of a trigonometric function is important because it helps us understand the behavior of the function near a specific value. It also allows us to determine if a function is continuous at a certain point, which is crucial in many real-world applications.

What are some common techniques for evaluating limits of trigonometric functions?

Some common techniques for evaluating limits of trigonometric functions include using the squeeze theorem, L'Hospital's rule, and the definition of a limit. You can also use special trigonometric limits, such as the limit of sin(x)/x as x approaches 0, to evaluate more complex functions.

Are there any restrictions when finding the limit of a trigonometric function?

Yes, there can be restrictions when finding the limit of a trigonometric function. For example, some functions may not have a limit at certain points or may have different limits from the left and right sides. It is important to consider these restrictions and analyze the behavior of the function to accurately determine the limit.

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