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student:mathematics:limits [2018/08/26 14:27] – created bernstdhstudent:mathematics:limits [2024/01/24 13:34] – bernstdh
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- +===== Limits =====
-===== Mathematical Foundations: Limits =====+
  
  
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 If you don't understand, an example should clear things right up. What happens to the fraction 1/n1/n as nn gets larger and larger?  Well, starting at n=1n=1 (and assuming that nn is an integer), as nn gets larger and larger we get a sequence of numbers 1/1,1/2,1/3,1/4,...1/1,1/2,1/3,1/4,....  Thus, what happens as nn gets larger and larger is that 1/n1/n gets smaller and smaller.  In fact, 1/n1/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/n1/n as nn gets larger and larger?  Well, starting at n=1n=1 (and assuming that nn is an integer), as nn gets larger and larger we get a sequence of numbers 1/1,1/2,1/3,1/4,...1/1,1/2,1/3,1/4,....  Thus, what happens as nn gets larger and larger is that 1/n1/n gets smaller and smaller.  In fact, 1/n1/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 \(n \rightarrow  \oo\) then 1/n→01/n→0 (i.e., if nn goes to infinity then 1/n1/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→01/n→0 (i.e., if nn goes to infinity then 1/n1/n goes to zero). Another way is to write:
  
-|   \(lim_{n \rightarrow  \oo} 1/n = 0\)   |+|   \(lim_{n \rightarrow  \infty} 1/n = 0\)   |
  
 which says that the limit of 1/n1/n as nn goes to infinity is zero. which says that the limit of 1/n1/n as nn goes to infinity is zero.
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   - Given two functions of xx, f1f1 and  f2f2, with finite limits, L1L1 and L2L2 it follows that limx→N(f1+f2)=L1+L2limx→N(f1+f2)=L1+L2   - Given two functions of xx, f1f1 and  f2f2, with finite limits, L1L1 and L2L2 it follows that limx→N(f1+f2)=L1+L2limx→N(f1+f2)=L1+L2
   - Given two functions of xx, f1f1 and  f2f2, with finite limits, L1L1 and L2L2 it follows that then limx→N(f1f2)=L1L2limx→N(f1f2)=L1L2   - Given two functions of xx, f1f1 and  f2f2, with finite limits, L1L1 and L2L2 it follows that then limx→N(f1f2)=L1L2limx→N(f1f2)=L1L2
-  - (L'Hopital's Rule) Given two differentiable functions of xx,  +  - (L'Hopital's Rule) Given two differentiable functions of xx, f1f1 and  f2f2, with infinite limits and derivatives \(f_{1}^{\prime}\) and \(f_{2}^{\prime}\), it follows that \(lim_{x \rightarrow \infty}(\frac{f_{1}}{f_{2}}) =  lim_{x \rightarrow \infty}(\frac{f_{1}^{\prime}}{f_{2}^{\prime}})\). 
-f1f1 and  f2f2, with infinite limits and derivatives f1′ and f2′, it follows that \(lim_{x \rightarrow \oo}(\frac{f_{1}}{f_{2}}) =  lim_{x \rightarrow \oo}(\frac{f_{1}\prime}{f_{2}\prime})\). +