Can anybody explain this to me?

[Artusartos]

In this link:


http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw9sum06.pdf

For number 28.8 b),

...for case 1, they say that x is the limit for the sequnce <x_n>, right? So doesn't the limit for the sequence <f(x_n)> need to be x^2? Why does the answer say that it must be zero?

 

[lurflurf]

That would be true for f continuous, which the exercise is proving to be false. That approach seems a bit complicated.
if x_n is always rational
lim <f(x_n)>=0
if x_n is never rational
lim <f(x_n)>=x^2

they agree if and only if x=0

 

[tiny-tim]

Hi Artusartos!

For number 28.8 b),

...for case 1, they say that x is the limit for the sequnce <x_n>, right? So doesn't the limit for the sequence <f(x_n)> need to be x^2? Why does the answer say that it must be zero?

No, it says lim_n->∞x_n² = x² ≠ 0.

(not equal to)