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May 22nd, 2009, 10:31 PM   #1
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need help on a question

Need urgent help... I have no clue how to do this; if anyone has an answer, I'd greatly appreciate it.

Q. Let F be the function such that F(x) = 1+x^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)

Fill in the blanks

Clearly, for all numbers x, F(x) >/= 1 (greater than or equal to 1). Next we note 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 $\in$ [____________, ____________]. Particularly, we get them for all ___(rational or non-rational)___ numbers if x in [ _____________,_____________] we have
F(x) = 1 + 4x^4 </= 1 + x^2

Therefore for all x $\in$ [_____________, _____________ ], we have that

1 </= F(x) </= 1+x^2

Since lim x-> 0 (1+x^2) = ______________, by the _____________ Theorem, lim x->0 F(x) = _________________.

Last edited by skipjack; February 7th, 2019 at 06:43 AM.
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May 23rd, 2009, 08:16 AM   #2
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Where are you stuck in following the given steps? You learned how to solving quadratic inequalities back in algebra, and you've learned in calculus about the Squeeze Theorem. At what point did you bog down?

Please be specific, showing all of your work so far. Thank you!
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May 23rd, 2009, 11:58 AM   #3
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Re: need help on a question

Sorry if I wasn't clear, but that is the exact sheet that my instructor gave me and told us to simply fill in the blanks... the reason why I'm asking for help is that I have no freaking clue what it's asking... thanks.

Last edited by skipjack; February 7th, 2019 at 05:46 AM.
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May 24th, 2009, 04:02 AM   #4
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Okay; then first follow the links (provided earlier) and learn how to solve quadratic inequalities and how the Squeeze Theorem works. (They were supposed to have taught you this material before assigning homework on it. Since you say they didn't, and since we cannot here provide the missing hours of classroom instruction, you'll have to attempt self-study.)

Once you have learned some of the necessary background material, please take another stab at the exercise. At the very least, you'll be able to do the algebra for solving the inequality. And that will help you fill in at least most of the blanks.

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February 6th, 2019, 10:16 PM   #5
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I suggest you look at the Limits Formula
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February 7th, 2019, 04:16 AM   #6
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The limit must be the same for the two functions .
Just solve the limits and it is 1 .
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February 7th, 2019, 06:17 AM   #7
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Given epsilon, you can determine delta st |F(x)-1| < epsilon when |x| < delta because x^2 and x^4 are arbitrarily small.
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February 7th, 2019, 07:08 AM   #8
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x²(1 - 4x²) $\small\geqslant$ 0 if and only if 1 - 4x² $\small\geqslant$ 0, which holds if and only if x $\in$ [-1/2, 1/2].
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