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student:mathematics:limits [2024/01/24 13:34] – bernstdh | student:mathematics:limits [2024/01/24 13:36] (current) – bernstdh |
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===== Limits ===== | ===== Limits ===== |
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If you don't understand, an example should clear things right up. What happens to the fraction \(1/n\) as \(n\) gets larger and larger? Well, starting at \(n=1\) (and assuming that \(n\) is an integer), as \(n\) gets larger and larger we get a sequence of numbers \(1/1, 1/2, 1/3, 1/4, ...\). Thus, what happens as \(n\) gets larger and larger is that \(1/n\) gets smaller and smaller. In fact, \(1/n\) gets closer and closer to zero. | If you don't understand, an example should clear things right up. What happens to the fraction \(1/n\) as \(n\) gets larger and larger? Well, starting at \(n=1\) (and assuming that \(n\) is an integer), as \(n\) gets larger and larger we get a sequence of numbers \(1/1, 1/2, 1/3, 1/4, ...\). Thus, what happens as \(n\) gets larger and larger is that \(1/n\) gets smaller and smaller. In fact, \(1/n\) gets closer and closer to zero. |
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Congratulations! You've just taken your first limit. How would you write this down? There are two ways. One way is to write: if \(n \rightarrow \infty \) then \(1/n \rightarrow 0\) (i.e., if \(n\) goes to infinity then \(1/n\) goes to zero). Another way is to write: | Congratulations! You've just taken your first limit. How would you write this down? There are two ways. One way is to write: if \(n \rightarrow \infty \) then \(1/n \rightarrow 0\) (i.e., if \(n\) grows without bound then \(1/n\) goes to zero). Another way is to write: |
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| \(lim_{n \rightarrow \infty} 1/n = 0\) | | | \(lim_{n \rightarrow \infty} 1/n = 0\) | |
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which says that the limit of \(1/n\) as \(n\) goes to infinity is zero. | which says that the limit of \(1/n\) as \(n\) goes to infinity (i.e., as \(n\) grows without bound) is zero. |
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So, how do you talk about situations where some number is getting really large or really small? You talk about what happens "in the limit". | So, how do you talk about situations where some number is getting really large or really small? You talk about what happens "in the limit". |