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Is there a way to use the limit definition to prove that the limit as $x$ approaches 0 of $\dfrac{\sin x} x = 1$? I get the definition (sort of), but it doesn't seem to have any practical applications in terms of extension of knowledge, just raw verification and mathematical rigor.

Thanks.

EDIT: John: it was a typo. Sorry for the confusion, the limit of sin(x)/x as x approaches infinity seems much more obvious. I am more interested as x approaches 0. My question really comes down to a much simpler concept: it seems like proofs often reveal to us new information through a series of logically consequent statements. However, I can't tell what the new information gained by epsilon delta proofs are, and I was wondering if there was some new mathematical idea that was revealed by this method.

louie mcconnell
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    No, because it's not true. It is true that that is the limit as $x \to 0$, however. Is that what you meant? The limit as $x \to \infty$ is $0$. – John Hughes Dec 19 '14 at 00:56
  • One can't say anything about limits without defining them. Without a definition, "limit" is a meaningless string of characters on a screen or paper. – GPerez Dec 19 '14 at 01:03
  • The idea of limit has not always been approached through the modern definition. Try searching online for the history of calculus and of the limit definition. This should give you some perspective on why the definition was developed and why people were not happy with earlier approaches. – David Dec 19 '14 at 01:03
  • Relative to David's comment, you could be asking why this particular definition is the one chosen for capturing the "notion" of a limit (which still has to be defined, even if roughly so). The answer to that lies in it's usefulness in shaping a solid ground for calculus and analysis. – GPerez Dec 19 '14 at 01:12
  • We know that the limit of both -1/x and 1/x as x approaches either positive or negative infinity is zero - first google definition, please search up your question first – Irrational Person Dec 19 '14 at 01:13
  • John: it was a typo. Sorry for the confusion, the limit of sin(x)/x as x approaches infinity seems much more obvious. My question really comes down to a much simpler concept: it seems like proofs often reveal to us new information through a series of logically consequent statements. However, I can't tell what the new information gained by epsilon delta proofs are, and I was wondering if there was some new mathematical idea that was revealed by this method. – louie mcconnell Dec 19 '14 at 01:38

2 Answers2

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That is incorrect, we have

$$\left| \frac{\sin x}{x} \right| \le \left| \frac{1}{x} \right| \rightarrow 0.$$

MathMajor
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This is very incorrect

Given

and

$$ f(x) = \frac{-1}{x}$$

$$ h(x) = \frac{1}{x} $$

since sin(x) is alway between range[-1, 1]

$$ f(x) \leq sin(x) \geq h(x)$$

concluding that for all x in $$(x, 0)$$

$$ \lim_{n \to 0} f(x) = 0$$

and

$$ \lim_{n \to 0} g(x) = 0$$

Concluding that

$$ \lim_{x \to 0} \frac{sin(x)}{x}= 0 $$

You can aslo try both negative and positive limits to find your answer. because we have proven that the limit as x approaches negative infinity or positive infinity is zero or that this limit doesn't exist.

Conclusion

since $$0\leq\sin(X)\geq0$$ We conclude the limit of sin(x) as x approaches 0 must be 0

Irrational Person
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