My proof for this limit using the definition is correct? #lim to 2^+ (1/(x-2)) = +\infty#

My answer:

For all A > 0, exists #\delta# > 0 such that:
#(1/(x-2)) > A# so that 0 < x+2 < #\delta#.

Looking on inequality bellow between B, we have the key choose for #\delta# :

#(1/(x-2)) > A#
#(x-2) < 1/A#
#x < 1/A+2#

Like this, for #\delta# = #1/A+2#, we have #1/(x-2) > A# always that 0 < x-2 < #delta#.

1 Answer
May 31, 2018

See explanation

Explanation:

There is one mistake: #0 < x+2 < delta#. After all, it is #x-2# that you want to go towards 0.

I might also want to refine the wording a little, for instance:
"For all A > 0, there exists a #delta>0# such that:" to make the proof clearer.

Also, as the proof presupposes that #x>2#, I might write:
#2<x<2+1/A# to make it clear that x lies between 2 and #2+1/A#.

One other detail: You introduce B, but it's not clear where B belongs or what it refers to.