(New page: <math>\text{True of False: If } f \text{ is a non-negative function defined on } \mathbf{R} \text{ and }</math> <math>\int_{\mathbf{R}}{f} dx < \infty </math> <math>\text{then } \lim_{|x...) |
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| − | + | [[MA_598R_pweigel_Summer_2009_Lecture_4]] | |
| − | <math>\int_{\ | + | <math>\text{4.1) True of False: If } f \text{ is a non-negative function defined on } \mathbb{R} \text{ and }</math> |
| + | |||
| + | <math>\int_{\mathbb{R}}{f} dx < \infty </math> | ||
<math>\text{then } \lim_{|x|\rightarrow\infty}f(x)=0</math> | <math>\text{then } \lim_{|x|\rightarrow\infty}f(x)=0</math> | ||
<math>\text{Solution: False. Let } | <math>\text{Solution: False. Let } | ||
| − | f(x)= \begin{cases} 1 & x\in \ | + | f(x)= \begin{cases} 1 & x\in \mathbb{Z} \\ |
0 & \text{otherwise}\end{cases} </math> | 0 & \text{otherwise}\end{cases} </math> | ||
| − | <math>\text{then } \int_{\ | + | <math>\text{then } \int_{\mathbb{R}}{f} dx = 0 \text{, but }\lim_{|x|\rightarrow\infty}f(x) \text{ does not exist.} </math> |
| + | |||
| + | -Ben Bartle | ||
Latest revision as of 18:10, 5 July 2009
MA_598R_pweigel_Summer_2009_Lecture_4
$ \text{4.1) True of False: If } f \text{ is a non-negative function defined on } \mathbb{R} \text{ and } $
$ \int_{\mathbb{R}}{f} dx < \infty $
$ \text{then } \lim_{|x|\rightarrow\infty}f(x)=0 $
$ \text{Solution: False. Let } f(x)= \begin{cases} 1 & x\in \mathbb{Z} \\ 0 & \text{otherwise}\end{cases} $
$ \text{then } \int_{\mathbb{R}}{f} dx = 0 \text{, but }\lim_{|x|\rightarrow\infty}f(x) \text{ does not exist.} $
-Ben Bartle
