Question

I have no clue on how to do this.. it says let f(x)= 3x. find the number delta > 0 so that 0<|x-2|< delta => |f(x)-6| < epsilon...

then it has a) epsilon=0.5 b)epsilon=0.1 c) epsilon=0.01 and d) epsilon is a (presumably small) positive number... :/

Answer 1

So epsilon is given in each problem, so you need to come up with a formula for delta in terms of epsilon.

So lets look at the expression:
\[|f(x)-6|<\epsilon \]what we want to do is make the left hand side look like |x-2|.

So doing a little bit of algebra we get:
\[|f(x)-6|<\epsilon \iff |3x-6|<\epsilon \iff 3|x-2|<\epsilon \]\[|x-2|<\frac{\epsilon}{3}\]
So let delta be epsilon over 3.

Answer 2

Ok ok.. so whats the next step? I kinda dont understand what im lookin for

Answer 3

So now we have a formula for delta in terms of epsilon. It is:
\[\delta =\frac{\epsilon}{3}\]
So for example, in a), the problem is, "find the number delta if epsilon is .5"
The correct delta would be:
\[\delta = \frac{.5}{3}=.16666\]

Answer 4

OMG thank you so much!