This seems not to have much to do with memoization, since as you say yourself, a new instance is started between those requests, so nothing would be 'stored'.
Still removing those duplicates in your code is a good idea. The Rails way to do this in your case would be a before_filter:
class ResetController < ApplicationController
before_filter :find_user
protected
def find_user
@user = User.find_by_email(params[:email])
# here you could add some exception handling and prevent execution of action if no user is found (assuming you want to do this)
end