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student:mathematics:limits [2018/08/26 14:27] – bernstdhstudent:mathematics:limits [2024/01/24 13:34] – bernstdh
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- +===== Limits =====
-===== Mathematical Foundations: Limits =====+
  
  
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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→∞ then 1/n→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→∞ then 1/n→0 (i.e., if n goes to infinity then 1/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/n as n goes to infinity is zero. which says that the limit of 1/n as n goes to infinity is zero.
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   - Given two functions of x, f1 and  f2, with finite limits, L1 and L2 it follows that limx→N(f1+f2)=L1+L2   - Given two functions of x, f1 and  f2, with finite limits, L1 and L2 it follows that limx→N(f1+f2)=L1+L2
   - Given two functions of x, f1 and  f2, with finite limits, L1 and L2 it follows that then limx→N(f1f2)=L1L2   - Given two functions of x, f1 and  f2, with finite limits, L1 and L2 it follows that then limx→N(f1f2)=L1L2
-  - (L'Hopital's Rule) Given two differentiable functions of x,  +  - (L'Hopital's Rule) Given two differentiable functions of x, f1 and  f2, with infinite limits and derivatives \(f_{1}^{\prime}\) and \(f_{2}^{\prime}\), it follows that \(lim_{x \rightarrow \infty}(\frac{f_{1}}{f_{2}}) =  lim_{x \rightarrow \infty}(\frac{f_{1}^{\prime}}{f_{2}^{\prime}})\). 
-f1 and  f2, with infinite limits and derivatives f1′ and f2′, it follows that \(lim_{x \rightarrow \oo}(\frac{f_{1}}{f_{2}}) =  lim_{x \rightarrow \oo}(\frac{f_{1}\prime}{f_{2}\prime})\). +