Is this an example? It doesn't sound like it fits the description of someone giving the character a task that they don't expect their target to complete. Here, the task-taker is taking on a mission herself, and then others are telling her it can't (or shouldn't) be done:
BlazBlue: Litchi Faye-Ling embarks on a quest to save her friend who got turned into an abomination called Arakune. Litchi has been at her task for a long time out of guilt of not preventing Arakune's transformation, and grief of losing him, a task that everyone that comes across her always says is impossible. It ends up as a deconstruction, as Litchi still attempts to selfishly do the impossible with just her friendship, even if she regretfully had to compromise her morals by joining anObviously Evil group of people. Nothing comes of it. She even goes along with a world-destroying plan to save Arakune, as it was her only option to save him.
In the middle of doing her task, she also genuinely reached out and befriended a kid with psychosis issues, Carl Clover. It was enough that Carl eventually stuck with her even when she's going along with the world-destroying plan, and by the end of it, one of them becomes the sidekick of the other as they put the one with the world-destroying plan (Carl's father, Relius) on their leash and possibly direct him to do what they like.
(temporarily removed from Video Games folder)
Is this an example? It doesn't sound like it fits the description of someone giving the character a task that they don't expect their target to complete. Here, the task-taker is taking on a mission herself, and then others are telling her it can't (or shouldn't) be done:
- BlazBlue: Litchi Faye-Ling embarks on a quest to save her friend who got turned into an abomination called Arakune. Litchi has been at her task for a long time out of guilt of not preventing Arakune's transformation, and grief of losing him, a task that everyone that comes across her always says is impossible. It ends up as a deconstruction, as Litchi still attempts to selfishly do the impossible with just her friendship, even if she regretfully had to compromise her morals by joining an Obviously Evil group of people. Nothing comes of it. She even goes along with a world-destroying plan to save Arakune, as it was her only option to save him.
- In the middle of doing her task, she also genuinely reached out and befriended a kid with psychosis issues, Carl Clover. It was enough that Carl eventually stuck with her even when she's going along with the world-destroying plan, and by the end of it, one of them becomes the sidekick of the other as they put the one with the world-destroying plan (Carl's father, Relius) on their leash and possibly direct him to do what they like.
I hope you get tiny bits of eggshell in all your omelettes for the rest of your life!