How do you prove that the limit of #5 - 2x= 1# as x approaches 2 using the epsilon delta proof?

Question

Cameron Alexander · Accepted Answer

Preliminary analysis

#lim_(xrarrcolor(green)(a))color(red)(f(x)) = color(blue)(L)# if and only if

for every #epsilon > 0#, there is a #delta > 0# such that: for all #x#, #" "# if #0 < abs(x-color(green)(a)) < delta#, then #abs(color(red)(f(x))-color(blue)(L)) < epsilon#.

So we want to make #abs(underbrace(color(red)((5-2x)))_(color(red)(f(x)) )-underbrace(color(blue)(1))_color(blue)(L))# less than some given #epsilon# and we control (through our control of #delta#) the size of #abs(x-underbrace(color(green)(2))_color(green)(a))#

Look at the thing we want to make small. We want to see the thing we control.

#abs((5-2x)-1) = abs(-2x+4) = abs (-2(x-2)) =abs(-2)abs(x-2) =2abs(x-2)#

And there's #abs(x-2)#, the thing we control

We can make #2abs(x-2) < epsilon# by making #abs(x-2) < epsilon/2#.

So we will choose #delta = epsilon/2#. (Any lesser #delta# would also work.)

(Detail: if #abs(x-2) < epsilon/2#, then we can multiply on both sides by the positive number #2# to get #2abs(x-2) < epsilon#.)

Now we need to actually write up the proof:

Proof

Given #epsilon > 0#, choose #delta = epsilon/2#. #" "# (note that #delta# is also positive).

Now for every #x# with #0 < abs(x-2) < delta#, we have

#abs(f(x)-1) = abs((5-2x) - 1) = abs(-2x+4)) = abs(-2)abs(x-2) = 2abs(x-2) < 2delta#

[Detail if #abs(x-2) < delta#, we can conclude that #2abs(x-2) < 2delta#. #" "# We usually do not mention this, but leave it to the reader. See below.]

And #2 delta = 2 epsilon/2 = epsilon#

Therefore, with this choice of delta, whenever #0 < abs(x-2) < delta#, we have #abs(f(x) - 1) < epsilon#

So, by the definition of limit, #lim_(xrarr2)(5-2x) = 1#.

We can condense a bit

for every #x# with #0 < abs(x-6) < delta#, we have

#abs(f(x) - 1) = abs((5-2x)-1)#

# = abs(-2(x-2))#

# = 2abs(x-2)#

# < 2delta = 2 epsilon/2 = epsilon#.

So, #abs(f(x) - 1) < epsilon#.

Samantha Adkison · Answer

undefined To prove that the limit of 5 - 2x = 1 as x approaches 2 using the epsilon-delta proof, we need to show that for any given epsilon greater than 0, there exists a corresponding delta greater than 0 such that if 0 < |x - 2| < delta, then |(5 - 2x) - 1| < epsilon. Let's proceed with the proof: Given epsilon > 0, we need to find a delta > 0 such that if 0 < |x - 2| < delta, then |(5 - 2x) - 1| < epsilon. First, let's manipulate the expression |(5 - 2x) - 1| < epsilon: |(5 - 2x) - 1| < epsilon |4 - 2x| < epsilon |-2(2 - x)| < epsilon 2|x - 2| < epsilon Now, we can see that if we choose delta = epsilon/2, then: 0 < |x - 2| < delta 0 < |x - 2| < epsilon/2 0 < 2|x - 2| < epsilon Therefore, we have shown that for any given epsilon > 0, there exists a corresponding delta > 0 (specifically, delta = epsilon/2) such that if 0 < |x - 2| < delta, then |(5 - 2x) - 1| < epsilon. Hence, we have proven that the limit of 5 - 2x = 1 as x approaches 2 using the epsilon-delta proof.