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terhorst
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#34 on the much-discussed http://ftp.ets.org/pub/gre/Math.pdf" :
Suppose [tex]f[/tex] is a differentiable function with [tex]\lim\limits_{x \to \infty }f(x)=K[/tex] and [tex]\lim\limits_{x \to \infty }f'(x)=L[/tex] for some [tex]K,L[/tex] finite. Which must be true?
Answer is 1. Is this because [tex]f[/tex] might be [tex]C^1[/tex]? Can you give an example of a function where the limit of the first derivative exists but the limit of the second derivative is not zero? Thanks!
Suppose [tex]f[/tex] is a differentiable function with [tex]\lim\limits_{x \to \infty }f(x)=K[/tex] and [tex]\lim\limits_{x \to \infty }f'(x)=L[/tex] for some [tex]K,L[/tex] finite. Which must be true?
- [tex]L=0[/tex]
- [tex]\lim\limits_{x \to \infty }f''(x)=0[/tex]
- [tex]K=L[/tex]
- [tex]f[/tex] is constant.
- [tex]f'[/tex] is constant.
Answer is 1. Is this because [tex]f[/tex] might be [tex]C^1[/tex]? Can you give an example of a function where the limit of the first derivative exists but the limit of the second derivative is not zero? Thanks!
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