What do Protestant churches teach about the fate of deceased infants? My question is posed to the Protestant churches that teach that man is saved by faith alone. 

If a person is saved by faith alone, what happens to children who die before they are able to conceive of the meaning of Christ's death and their own need for him and therefore cannot have a saving faith in his work?

I heard the John Calvin taught that unbaptized infants were saved by their parents' faith. I've also heard that some Calvinist pastors will say that there is no way of knowing, because God has chosen which will go to heaven and hell apart from human action. Some Roman Catholic teachers taught that there was a place in Hell (or between Heaven and Hell) called limbo for unbaptized infants, but I see that Pope Francis has stated that unbaptized infants do have hope of salvation (http://www.telegraph.co.uk/news/worldnews/1549439/The-Pope-ends-state-of-limbo-after-800-years.html). But I'm looking for an overview of the positions that are taught as doctrine in the Protestant churches, or are at least in writing from their leaders.
The salvation of infants within Protestant denominations cover a wide range of alternate views.  The subject is closely related to different view of infant baptism but not identical. To avoid the subject of how baptism has its role in the subject, I would like to limit the scenario to a child that dies before having a chance of being baptized.

To keep things simple I would like to first list the main concepts involved that divide the views depending on the persons emphasis or belief in the principle ideas.


Virtually all protestants believe in original sin, therefore infants do need salvation.
Some believe that God can and does infuse a germ of faith into the soul of an infant at any age, regardless of the child's disposition or own will through irresistible grace. (This would be more common with Luther and Calvin and is similar to regenerating baptism)
Some believe in point (2) above but only for the elect. (This would be more common among Calvinistic Reformed Churches who put an emphasis on election. Some Baptists might lean in this direction also.)
Some believe faith is not even required for those assumed too young to sin of their own conscious choice and that faith is only required in those who are old enough to sin on their own accord. In other words everyone receives Christ until they reject Him. (This can be popular among almost any evangelical churches including Baptist, possibly less frequent in the traditional Lutheran)
Some believe (3) but that faith is not infused in the infant but counted for what God could foresee if the child were to have lived. In this sense there is allowance of what would have been under different conditions which might possibly be part of God's hidden reasons for the election in the first place. However, strictly speaking most classical Calvinists would oppose that idea for anything remotely implying that God's election has anything to do with man must be excluded. Therefore they revert back to simple infused irresistible grace. Yet, nevertheless even hyper Calvinists might hold this belief that God's foreknowledge of different circumstances 'may' be involved as they can't see how God's knowledge can be excluded from any of his decisions. Pushing the idea of foreknowledge towards an actual full reason for election is where Armenians and Methodists are usually found. 
Some believe in point (2) but not just for the elect but practically speaking for those children of parents who are believers that baptize their children as infants. (I know I said I would exclude baptism but just as a branch of common ground I raise this one concept as virtually ceremonializing who is elect with respect to children.) This would be common in traditional and somewhat ceremonial churches like some Lutheran, some Anglican, I think some Presbyterian but am not sure. Some traditional reformed churches also I think might lean in this direction. The other varient of this of course is a mixture with (4). Under the mixed view, once an infant is too old, the security of baptism might 'wear off' so to speak and confirmation is needed.  The reason is that an age of innocence has passed, whereby their were protected from their parents faith and baptism and now must believe or reject the gospel for themselves.


Why so many views? Because the Bible does not directly deal with the question or the answer. It is to be expected that where the Bible does not directly lay out a specific view directed to the subject of infant deaths, or at least not do so in an obvious way, naturally various different views will develop. The bible puts its focus on those who are able to choose and believe, or reject the offer of eternal life. If this makes someone feel frustrated it might be worth noticing that we really can't know for sure who are genuine Christians or not anyway. We can only obtain assurance for our own soul for our own condition. So if a child dies at 1 month or 40 years, we can only hope for their soul anyway. Our faith and hope must be placed in God, both for ourselves and for others.

References
I have managed to find a good reference that seems to more or less categorize the different Protestant beliefs as I have done. All quotes from Baker Encyclopedia of Christian Apologetics, Norman L. Geisler, P360-365)

Baptized Infants Only


  This view is held by sacramentalists, who believe that baptism is necessary for salvation. Some Roman Catholics, some Lutherans, and Anglicans espouse the position.


Elect Infants


  The elect infant view has not found a home outside of very strong Calvinistic circles. 


Those God “Foreknows.”


  According to this position, God, as an omniscient Being, foreknew which infants would have believed if they had lived long enough. 


All Infants


  Since the seventeenth century the view that all infants are saved has become the most popular in varying theological traditions. Some believe that all infants will eventually believe. Others believe that God will save infants apart from the condition that they would believe.