Warning: the rest of this post will be mainly literature-related.
In the first book of Madeleine L'Engle's Time Quintet, A Wrinkle in Time, the main character, Meg, is faced with an evil intelligence that is holding her brother captive. Meg has to go back alone to rescue her brother from IT, much to her own dismay. Mrs. Who, Mrs. Whatsit, and Mrs. Which, angelic beings that guide her throughout the book, advise her to look for what she has that IT does not: Love. As she stands before IT, Meg realizes that
"If she could give love to IT perhaps it would shrivel up and die, for she was sure that IT could not withstand love. But she, in all of her weakness and foolishness and baseness and nothingness, was incapable of loving IT. Perhaps it was not too much to ask of her, but she could not do it."
Instead, Meg loves her brother, though he has been distorted by IT, and she saves him. But in the second book she is faced with the same problem. The Ecthroi, of which IT was only a part, again threaten her brother. And this time loving him isn't enough. She has to love the Ecthroi. Though she is still weak and foolish and base and unqualified, she has to do it.
In The Brothers Karamazov, Dostoyevsky's character Ivan says that "One can love one's neighbors in the abstract, or even at a distance, but at close quarters it's almost impossible." He tries to make up for this by loving an abstract ideal of humanity and children, who he sees as pure and innocent, which proves that he knows nothing about children. The thing is, though, that he's right. Our sin makes us ugly, contemptible, impossibly unlovely. And yet God calls us to love each other. Out of the humanly impossible, God demonstrates his power by loving something that is utterly undeserving. This is what he calls us to do, and he knows that we can't do it without him. Because of our sin, we can't even do it as Christians, not 100% of the time. We're going to fail, no question--but we have to try anyway.
A priest in The Brothers Karamazov says, "Can there be a sin which could exceed the love of God?...Believe that God loves you as you cannot conceive; that he loves you with your sin, in your sin....All things are atoned for, all things are saved by love. If I, a sinner, even as you are, am tender with you and have pity on you, how much more will God." So here is the two-part answer to the question of why God calls us to do what is impossible. First of all, it shows us over and over again our deep, unending need of him, and his faithfulness and love for us. Secondly, it makes us present the image of Christ more significantly than ever, in demonstration and comparison. As people are unlovely, our love, such as it is and aided by the example of Christ, can a powerful witness to the extravagant love of God.
Love Through Me
Love through me, Love of God;
That burns perpetually.
Flow through me, Peace of God;
Calm River, flow until
No wind can blow, no current stir
A ripple of self-will.
Shine through me, Joy of God;
Make me like Thy clear air
That Thou dost pour Thy colors through,
As though it were not there.
O blessed Love of God,
That all may taste and see
How good Thou art, once more I pray:
Love through me—even me.