How do you prove the statement lim as x approaches 1 for #(5x^2)=5# using the epsilon and delta definition?
See the explanation section below.
We need to make
So we begin by examining
# = 5 abs((x+1)(x-1)) = 5 abs(x+1) abs(x-1)#
We control, through our choice of
If we make sure that
(If we put a bound on the distance between
Any number will do, but is might be easiest for purpose of illustration to pick a number we haven't use yet so we can keep track of it.
Let's make sure that
So, adding 2 to each part, we get,
#"which is" < 5(4)abs(x-1)#
# "which is"= 20 abs(x-1)#
#"we want this" < epsilon#
So let's make sure that, in addition to
we also want
Now we are ready to write the proof.
note first that
Furthermore, for such
# < 5(4) abs(x-1) = 20abs(x-1)#
# < 20(epsilon/20) = epsilon#.
That is: for
Therefore, by the definition on limit,
# lim_(xrarr1)5x^2 = 5#