In Jane Austen's Persuasion and Fitzgerald's The Great Gatsby, how is love seen as fragile and easily destroyed?

Although Gatsby ends tragically and Persuasion ends happily, we can make a strong case that "love is fragile" is a reality that is portrayed by both books.  Both books show love persisting for a long time, over an absence, but they also show how it can fail because of seemingly minor factors.

In Persuasion, Anne Elliot continues loving Captain Wentworth even after she breaks off her engagement to him on the advice of her family.  She still has feelings for him after seven years of not seeing him.

In Gatsby, Jay Gatsby continues to love Daisy for five years after she marries another man.  He goes to the extent of making a fortune and buying a house across the bay from hers, hoping to win her back. 

In the case of Jay Gatsby, he imagines that all he has to do is reunite with Daisy and she will immediately declare her love for him and leave her husband and daughter.  It turns out that he has overestimated Daisy's strength of character, or underestimated her practicality.  Although she loves Gatsby, she does not want to disturb her secure lifestyle.  When Gatsby finally forces a showdown between himself and Daisy's husband Tom, the self-confident Tom is able to confuse and distract Daisy by exposing Gatsby as a fake.  Daisy simply runs out of emotional energy to go through with leaving Tom.  She would have to give up so much, that even a small amount of confusion is enough to deflect her purpose.  When Gatsby is killed the next day, she does not even attend his funeral.

Persuasion is more complicated.  Wentworth is understandably disillusioned with Anne after she breaks off the engagement.  While Gatsby's naive idea of Daisy led him to overestimate her confidence and maturity, Wentworth mistakenly concludes that Anne's choice to follow her family's advice shows that she is weak in character.  In fact, Anne has matured over the years since she last saw Wentworth.  Wentworth eventually recognizes this, and proposes to Anne again.  

This might look like it is about the persistence of love.  But in the interval, Wentworth was starting to become involved with another young woman, Louisa Musgrove.  He starts courting her, telling her that he values "determination" above all else, and obviously hoping that she is more "determined" than Anne and more likely to be true to him.  But he slowly starts to realize that she is just an average, immature young lady (just like Anne once was).  He still seems interested in her; then, when jumping from some stairs to be caught by him, she hits her head in a fall, is concussed, and spends some time in a coma.  This is all it takes for Wentworth to gracefully drop her.  Everyone is surprised when Wentworth does not go to visit Louisa on her sickbed, but it slowly becomes clear that he does not intend to pursue the relationship any further.

So, in Persuasion we have Anne dropping Wentworth simply because she is young, unsure, and easily influenced by her family.  Then, we have Wentworth losing interest in Louisa over something that isn't really her fault.  (Neither the accident nor the fact of being young and immature are really her fault.)  However, the actual reason he loses interest in her is perhaps that he realizes he was a bit too hard on Anne.  

Persuasion does not really show that love itself is fragile, but it does show how the smallest details of circumstances and bad timing may derail a love affair.  But as Anne points out, just because things do not work out between two people does not mean that they don't continue to love each other.

The message is almost the opposite in Gatsby: Just because things do work out, doesn't necessarily mean that love is present.  Gatsby and Daisy had an affair and they still feel a spark, but Gatsby was more in love with Daisy's way of life than with the actual woman, and Daisy's attraction to Gatsby did not mean that she loved him enough to sacrifice anything for him. 

All in all, Persuasion is about how society and circumstances may conspire against those who love each other, whereas Gatsby is about how lovers might be undone by their blind spots and their own treacherous hearts.