Limit of sin(3x)/sin(5x) as x→0

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In summary, the conversation discusses finding the limit of the function (sin(3x))/(sin(5x)) as x approaches 0. The solution is found to be 0.6 by using a calculator, but the process of proving it is unknown. The suggestion of multiplying by certain values and using the limit rule for (sin u)/u is given as a possible approach. A related thread with a similar problem is mentioned.
  • #1
PirateFan308
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Homework Statement


Find [tex]\lim_{x\rightarrow 0} {\frac{\sin(3x)}{\sin(5x)}}[/tex]


The Attempt at a Solution


I know that the limit equals 0.6 (by typing it into my calculator), but I have no idea how to prove this, or even where to start. I know that sin is continuous, so I theoretically should be able to just plug it in, but obviously this doesn't work because it isn't divisible by 0.
 
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  • #2
Try multiplying by 3x/(3x) and 5x/(5x), and placing the numerators and denominators strategically. The basic idea is that [itex]\lim_{u \to 0} \frac{sin u}{u} = 1[/itex]
 
  • #4
Thank you! I can't believe I didn't think of that answer - that helped me figure out the later questions as well.
 

1. What is the limit of sin(3x)/sin(5x) as x approaches 0?

The limit of sin(3x)/sin(5x) as x approaches 0 is 3/5.

2. How do you determine the limit of a trigonometric function?

To determine the limit of a trigonometric function, we can use the basic trigonometric identities and properties, as well as the concept of continuity. In this specific case, we can use the limit laws and the fact that sin(x) is continuous at x = 0 to evaluate the limit of sin(3x)/sin(5x) as x approaches 0.

3. Can the limit of sin(3x)/sin(5x) as x approaches 0 be evaluated using L'Hopital's rule?

Yes, L'Hopital's rule can be used to evaluate the limit of sin(3x)/sin(5x) as x approaches 0. However, it may be more efficient to use the basic trigonometric identities and properties.

4. What is the significance of the limit of sin(3x)/sin(5x) as x approaches 0?

The limit of sin(3x)/sin(5x) as x approaches 0 is the slope of the tangent line to the graph of the function at x = 0. It also represents the instantaneous rate of change of the function at that point.

5. Are there any restrictions on the values of x for which the limit of sin(3x)/sin(5x) exists?

No, there are no restrictions on the values of x for which the limit of sin(3x)/sin(5x) exists. However, the function is undefined at x = 0, as dividing by 0 is not allowed. This is why we are evaluating the limit as x approaches 0 instead of plugging in x = 0 directly.

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