Limitation of Unemployment rate What is the rationale behind government not including discouraged workers who have given up looking for jobs out of frustration given the state of economy?

This under-reports the job market and at the outset looks ridiculous.
The U-3 unemployment rates is the total unemployed, as a percent of the civilian labor force (official unemployment rate). By definition it excludes the discouraged workers, which, as Varun and Kent understand, is not true of all unemployed statistics. I liked what Karabell (2014) says about this statistic:


  ...[T]here is nothing wrong per se with the [U-3] unemployment figure,
  provided you know that the definition of employment is a construct,
  not a counting exercise....to be unemployed, according to the
  Bureau of Labor Statistics, you have to be in the workforce. To be in
  the workforce, you have to have a job or be looking for a job. To be
  looking for a job, you have to answer when surveyed that you have
  actively sought employment at some point in the past four weeks. If
  you haven't, you aren't actually unemployed. You are a discouraged
  worker, are marginally attached to the workforce, and are not counted
  as part of said workforce. And if you haven't been looking for a year,
  you're not, statistically speaking, even discouraged. You simply,
  statistically speaking, don't exist. You do, of course, actually exist
  (though that is a whole other topic), but not in the statistical
  universe of employment data. 
  
  The result of such methods is that it is possible, and not just
  theoretically, for the unemployment rate to drop not because more jobs
  were created or more people were hired but simply because more people
  gave up looking for work and dropped out of the workforce. That is, in
  fact, What happened in the United States and swaths of Europe after
  2009, though in America, there were also real job gains in 2010 and
  after. Nonetheless, the unemployment rate, which receives so much
  public scrutiny, went down in part not because the economic system was
  creating so many new jobs but because it was failing to do just that.
  To be fair, the BLS keeps multiple unemployment statistics and
  different unemployment rates, including ones that include
  underemployed, marginally attached, and part-time workers, but there
  is only one headline number. That is what the BLS points to in its
  monthly release; that is what the media pick up; and the other numbers
  are at best footnotes.


The Leading Indicators: A SHort History of the Numbers That Rule Our World
Zachary Karabell (2014)

Therefore, if your question is why does U-3 exclude discouraged workers, the answer is that's an essential part of the definition of discouraged. 

If your question is why did they define U-3 this way in the first place, I am not sure. I suspect the following. The definition of the labor force, which is workers and those actively looking for work or waiting to be recalled to work. Detached and marginally attached workers are not in the labor force. Once you defined the labor force, a natural question is what fraction of the labor force is not working, and that statistic is U-3. 

If your question is why is this statistic more widely discussed than other economic statistics, the answer may simply be that it is older. The U-3 statistic goes back to 1948 while U-4, U-5, and U-6 start in 1994. People may be used to talking about the better established statistic and enjoy the longer time series, and see insufficient benefit to switching to the newer statistics, despite the benefits. A sort of Dvorak vs. Qwerty keyboard issue for economic statistics. 

In a way, this is like an issue of GDP measurement. Home production is not in GDP and Robert Barro has argued that investment is double counted in GDP. If you want to use GDP as a measure of something else, these may be big problems and other measures may serve better. However, if you want to compare GDP across countries or time, these issues matter less than having something useful and consistent to compare them with. Commenter ABC at Marginal revolution has a nice quote about this:


  The algebra is fine as it is, but I take strong issue with the idea
  that GDP "mismeasures" something. As any decent intro macro course
  will tells its students, "GDP is not a measure of welfare." Indeed,
  the Kuznets quotes that Barro points to are exactly of this nature; we
  have this high profile GDP number, but it doesn't do what you want it
  to do - or at least not ALL the things you want it to.
  
  What Barro does here is construct a new measure - present discounted
  value of consumption - and shown how it relates to GDP (and GDP's
  present discounted value). In addition, he provides a capital income /
  labor income decomposition for both measures (GDP having such an exact
  decomposition because of constant returns to scale, Euler's identity
  for homogenous functions, and marginal-value input prices). He then
  says present discounted value of consumption, and its associated
  decompositions, are "right" while the other the construction GDP is
  "wrong" and thus gives "misstated" decompositions. Of course,
  "right/wrong" and "correct/incorrect" begs the question - right about
  WHAT? Correct for WHAT?
  
  Barro says that in present discounted value terms GDP "double counts"
  investment. The "double counting" is only relative to how things are
  calculated for the present discounted value of consumption. This just
  reflects the fact that GDP counts production, while consumption only
  counts consumption.


ABC's comment on Marginal Revolution Post: Robert Barro says we double-count investment