Question

which of the following statements are true for all values of c?

I. lim f(x)=0 =>lim |f(x)|=0
x->c x->c

II. lim |f(x)|=0 =>lim f(x)=0
x->c x->c
are they both true? both false, or is either one true but the other false? HELP!

Answer 1

got me thinking, but i would say true. not true if it wasn't zero, but since it is zero i think it is right
johnny? i would love to see a counterexample

Answer 2

so you would say both are true?

Answer 3

oh silly me we go from the definition.

Answer 4

suppose f \[\mathbb{R}-\mathbb{R}\] C and 0 are \[\epsilon \mathbb{R}\] then there exists \[\delta>0 \]

Answer 5

what does it mean to say
\[\lim_{x\rightarrow c}f(x)=L\]? it means given
\[\epsilon > 0\] there is a
\[\delta >0\] such that if
\[|x-c|<\delta\]
\[|f(x)-L|<\epsilon\]

Answer 6

replace L by 0 and i believe you have what you need

Answer 7

beat me to the punch, fast at typing haha

Answer 8

ok will do.

Answer 9

hmm maybe there is something to the second one though?

Answer 10

no there isn't

Answer 11

so 1 is true while 2 is false

Answer 12

its been awhile since ive used the limit theorem, reminds me of undergrad, proving triangle inequality

Answer 13

I just didn't know the rules as they applied to absolute values because we never really went over that.

Answer 14

i will say they are both true. they say the same thing