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Is the statement "if the limit of f(x) as x approaches c is 0, then there must exist a number k such that f(k)<0.001" a true or false statement?

Answer 1

It is true.

If #lim_(xrarrc)f(x) = 0#, the for any positive #epsilon#, (like #0.001#) there is a positive #delta# such that for every #x# with #0 < abs(x-c) < delta# we have #abs(f(x)-0) < epsilon#.

There is not only one value of #k#, but for the #delta# that works for #epsilon = 0.001#, every #x# in #(c-delta, c+delta)# has #absx < 0.001#, so it has #f(x) < 0.001#

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Answer 2

Luke Alvarez

False.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Is the statement "if the limit of f(x) as x approaches c is 0, then there must exist a number k such that f(k)&lt;0.001" a true or false statement?

Is the statement "if the limit of f(x) as x approaches c is 0, then there must exist a number k such that f(k)<0.001" a true or false statement?