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student:mathematics:limits [2018/08/30 13:21] – bernstdhstudent: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.
  
-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:
  
 |   \(lim_{n \rightarrow  \infty} 1/n = 0\)   | |   \(lim_{n \rightarrow  \infty} 1/n = 0\)   |
  
-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.
  
 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".
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   - Given two functions of \(x\), \(f_{1}\) and  \(f_{2}\), with finite limits, \(L_{1}\) and \(L_{2}\) it follows that \(lim_{x \rightarrow N}(f_{1} + f_{2}) = L_{1} + L_{2}\)   - Given two functions of \(x\), \(f_{1}\) and  \(f_{2}\), with finite limits, \(L_{1}\) and \(L_{2}\) it follows that \(lim_{x \rightarrow N}(f_{1} + f_{2}) = L_{1} + L_{2}\)
   - Given two functions of \(x\), \(f_{1}\) and  \(f_{2}\), with finite limits, \(L_{1}\) and \(L_{2}\) it follows that then \(lim_{x \rightarrow N}(f_{1} f_{2}) = L_{1} L_{2}\)   - Given two functions of \(x\), \(f_{1}\) and  \(f_{2}\), with finite limits, \(L_{1}\) and \(L_{2}\) it follows that then \(lim_{x \rightarrow N}(f_{1} f_{2}) = L_{1} L_{2}\)
-  - (L'Hopital's Rule) Given two differentiable functions of \(x\),  +  - (L'Hopital's Rule) Given two differentiable functions of \(x\), \(f_{1}\) and  \(f_{2}\), with infinite limits and derivatives \(f_{1}^{\prime}\) and \(f_{2}^{\prime}\), it follows that \(lim_{x \rightarrow \infty}(\frac{f_{1}}{f_{2}}) =  lim_{x \rightarrow \infty}(\frac{f_{1}^{\prime}}{f_{2}^{\prime}})\). 
-\(f_{1}\) and  \(f_{2}\), with infinite limits and derivatives \(f_{1}^{\prime}\) and \(f_{2}^{\prime}\), it follows that \(lim_{x \rightarrow \infty}(\frac{f_{1}}{f_{2}}) =  lim_{x \rightarrow \infty}(\frac{f_{1}^{\prime}}{f_{2}^{\prime}})\). +