Proving Limit of (sin 5x)/x: Squeeze Theorem

• ktpr2
In summary, the conversation is discussing the limit of g(x) as x approaches 0 and how to prove that f(x) is less than g(x) which is less than h(x). The squeeze theorem is mentioned as a possible method for proving this and there is also a suggestion to use the fact that the limit of sin x divided by x is 1. The conversation also references a thread on a similar limit in the math section for further ideas.
ktpr2
$$f(x) = -50x^2+5$$
$$g(x) = (sin 5x)/x$$
$$h(x) = x^2+5$$

I'm trying to find the limit of g(x) as x --> 0

I know that f(x) and h(x) are less than and greater than, respectively, than g(x) but I am unsure how to prove that w/o abusing the concept of infinity. How would I prove this so that i can show that

$$f(x)\leqq g(x) \leqq h(x)$$

and use the squeeze theorem to show that the limit as x --> 0 for g(x) = 5 because it's also 5 for f(x) and h(x)? Alternatively, I'm sure, is there a better way to go about this?

$$\lim_{x \rightarrow 0} \frac{\sin (5x)}{x} = \left(\frac{d}{dx} \sin (5x)\right) \biggr |_{x =0} = 5\cos{0} = 5.$$

Of course, you have to know a bunch of things to be able to use that to start with. Another easy way, if you already know

$$\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$$

then just multiply top and bottom of your limit by $5$ and note that $x\rightarrow 0 \Longleftrightarrow 5x \rightarrow 0$.

There's a thread on a similar limit in the math section that might give you some ideas (there's a geometrically motivated proof there too, using the squeeze theorem, if you want to do it that way):

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The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) for all values of x near a (except possibly at a), and if lim f(x) = lim h(x) = L, then lim g(x) = L as x → a.

In this case, we can see that f(x) = -50x^2+5 is always less than g(x) = (sin 5x)/x for all values of x near 0 (except possibly at 0), since -50x^2+5 is a quadratic function that approaches 0 as x approaches 0, while (sin 5x)/x is a trigonometric function that oscillates between -1 and 1 as x approaches 0. Similarly, h(x) = x^2+5 is always greater than g(x) for all values of x near 0 (except possibly at 0), since x^2+5 is a quadratic function that approaches 5 as x approaches 0, while (sin 5x)/x oscillates between -1 and 1.

Therefore, we can conclude that -50x^2+5 ≤ (sin 5x)/x ≤ x^2+5 for all values of x near 0 (except possibly at 0). And since both -50x^2+5 and x^2+5 approach 5 as x approaches 0, we can use the Squeeze Theorem to show that the limit of g(x) as x approaches 0 is also equal to 5. This means that the limit of g(x) as x approaches 0 is indeed 5, and we have successfully proven it using the Squeeze Theorem.

In summary, the Squeeze Theorem is a powerful tool that allows us to find the limit of a function by "squeezing" it between two other functions whose limits we already know. In this case, we used the Squeeze Theorem to show that the limit of g(x) as x approaches 0 is 5 by showing that it is always between -50x^2+5 and x^2+5, both of which have a limit of 5 as x approaches 0. This is a valid and rigorous way to prove the limit without "abusing" the concept of infinity.

1. What is the Squeeze Theorem?

The Squeeze Theorem, also known as the Sandwich Theorem, is a mathematical theorem that helps to evaluate the limit of a function by comparing it to two other functions that are known to have the same limit. It is commonly used to prove the limit of trigonometric functions, such as the limit of (sin 5x)/x.

2. How is the Squeeze Theorem applied to prove the limit of (sin 5x)/x?

In order to prove the limit of (sin 5x)/x using the Squeeze Theorem, we must find two other functions that "squeeze" the original function and have the same limit. These functions should be simpler to evaluate and have known limits. In this case, we can use the functions 1 and -1 as our "squeezing" functions, as they have the same limit of 0 as x approaches 0. By comparing (sin 5x)/x to these two functions, we can prove that the limit of (sin 5x)/x also approaches 0 as x approaches 0.

3. Why is the Squeeze Theorem important in calculus?

The Squeeze Theorem is important in calculus because it provides a useful tool for evaluating limits of functions that may be difficult to evaluate directly. It allows us to use simpler, known functions to prove the limit of a more complex function. In addition, it is a fundamental theorem that is used in many other mathematical proofs and applications.

4. Can the Squeeze Theorem be used for other types of functions?

Yes, the Squeeze Theorem can be used for other types of functions, not just trigonometric functions. It can be applied to any function that can be "squeezed" between two other functions with the same limit. This makes it a versatile tool in calculus and can help in evaluating limits of various types of functions.

5. Are there any limitations to using the Squeeze Theorem?

While the Squeeze Theorem is a powerful tool, it does have some limitations. One limitation is that it can only be used to evaluate the limit as x approaches a specific value, such as 0 or infinity. It cannot be used to evaluate limits at other points, such as x = 1. In addition, finding suitable "squeezing" functions can sometimes be challenging and may require some trial and error.

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