Piecewise functions and their limits

  • Thread starter maiad
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  • #1
maiad
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Homework Statement


Let f be the function such that f(x)=1+^2 if x is irrational and f(x)=1+4x^4 if x is rational.

Use the Squeeze Theorem to find (lim x->0)f(x).
Clearly, for all real numbers x, f(x)>=1. Next, we note that for a real number x,

x^2−4x^4=(x^2)(1−4x^2)>=0
if and only if ? >=0. This means that x^2>=4x^4 if and only if x [?,?]

The Attempt at a Solution


I've been stuck on this question for a while.any hints to start me off?
 

Answers and Replies

  • #2
HallsofIvy
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If [itex]x_n[/itex] is a sequence of rational numbers, converging to 0, what is [itex]f(x_n)[/itex] for all n? What is the limit as n goes to infinity?

If [itex]x_n[/itex] is a sequence of irrational numbers, converging to 0, what is [itex]f(x_n)[/itex] for all n? What is the limit as n goes to infinity?
 

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