How Do You Calculate the Limit of 3sin(5x)/sin(3x) as x Approaches 0?

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In summary: In other words, the limit is \lim_{x\to0}3\frac{\sin(5x)}{(\sin(2x))}, or \lim_{x\to0}7.5.In summary, the function you are trying to solve is in the form \lim_{x\to0}x^3\frac{\sin(5x)/x}{\sin(3x)/x}, where the limits are found by taking the limits of the numerator and denominator separately. However, the limit you are trying to find is not 7.5, it is 7.5 - \lim_{x\to0}3\frac{\sin(5x)}{\sin(
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Aychetown
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I was on khan academy and I'm beginning to relearn calculus for the first time in years and I couldn't figure out how to prove this function. I think I was supposed to use my calculator but I would like to know if anyone can work it out for me?

It was the limit as x-0 of 3sin(5x)/sin(3x). The answer is supposed to be 7.5 but I can't figure out how to work it out. Help me my fellow math heads.
 
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Aychetown said:
I was on khan academy and I'm beginning to relearn calculus for the first time in years and I couldn't figure out how to prove this function. I think I was supposed to use my calculator but I would like to know if anyone can work it out for me?

It was the limit as x-0 of 3sin(5x)/sin(3x). The answer is supposed to be 7.5 but I can't figure out how to work it out. Help me my fellow math heads.
Hi Aychetown and welcome to MHB!

In this problem, you are probably meant to start from the fact that \(\displaystyle \lim_{x\to0}\frac{\sin x}x = 1.\) If you replace $x$ by $5x$, you get \(\displaystyle \lim_{x\to0}\frac{\sin (5x)}{5x} = 1.\) And if you then multiply that by $5$, it becomes \(\displaystyle \lim_{x\to0}\frac{\sin (5x)}{x} = 5\) (and the same thing would work if you replace the $5$ by any other multiple).

Now suppose you write your limit as \(\displaystyle \lim_{x\to0}3\frac{\sin(5x)/x}{\sin(3x)/x}\) (dividing top and bottom by $x$). You can then take the limits of the numerator and denominator separately, to get the answer. But the answer will not be 7.5, because you seem to have stated the problem incorrectly. My guess is that it should be \(\displaystyle \lim_{x\to0}3\frac{\sin(5x)}{\sin(2x)}\), with a $2$ in the denominator.
 

FAQ: How Do You Calculate the Limit of 3sin(5x)/sin(3x) as x Approaches 0?

1. How do you prove a simple function?

To prove a simple function, you need to show that it satisfies the definition of a function, which is that each input has only one corresponding output. You can do this by using a direct proof, where you start with the definition and use logical reasoning to show that the function satisfies it.

2. What is the first step in proving a simple function?

The first step in proving a simple function is to clearly define the function and its domain and codomain. This will help in understanding what the function is doing and what its intended outputs are.

3. What are the common methods used to prove a simple function?

The most common methods used to prove a simple function are direct proof, proof by contradiction, and proof by induction. Direct proof involves starting with the definition and using logical reasoning to show that the function satisfies it. Proof by contradiction involves assuming that the function does not satisfy the definition and then showing that this leads to a contradiction. Proof by induction involves proving the function for the base case and then showing that if the function works for n, it also works for n+1.

4. Can you provide an example of proving a simple function?

Sure, let's prove that the function f(x) = x + 2 is a simple function. We start with the definition that each input has only one corresponding output. If we choose any two inputs, say x = 3 and x = 4, we can see that the corresponding outputs are f(3) = 5 and f(4) = 6. This shows that each input has only one corresponding output, satisfying the definition of a function. Therefore, f(x) = x + 2 is a simple function.

5. How do you know if a simple function has been proven correctly?

A simple function is considered proven correctly if it satisfies the definition of a function and if the proof is logically sound. This means that the steps taken in the proof are valid and that the conclusion follows logically from the premises. It is important to carefully check each step of the proof to ensure its validity.

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