I may give you an answer : "à" is linked with an objective, a goal : 8. "Ce passe-partout sert à entrer dans n'importe quelle pièce en cas de besoin." -> the goal is to enter everywhere. (1. "Il faut que je pense à rapporter ce livre à la bibliothèque." -> here "penser" is just often used with "à") Other examples : "Il cherche à me voir." -> "He is trying to see me." or "He wants to see me.", the objective is to see me. "J'arrive à allumer la bougie." -> the goal is to light up the candle "de" means "of" in fact, just ask "of what ?", "to what ?" : 3. "Pourquoi n'as-tu pas envie de lire ce roman ?" -> "envie DE quoi ?" ; "want what ?" ; there is no goal actually 2. "J'ai besoin d'apprendre vos coutumes avant de venir vous voir." -> "besoin DE quoi ?" ; "need what ?" ; there is no goal 7. "Je viens d'apprendre ton accident." -> "je viens DE quoi ?" ; "you just what ?" ; no goal again etc. I know it's hard. But you can see a little difference because there is in the preposition "à" an idea of a goal, an objective, something to realize. The preposition "de" is juste used to specify, indicate, precise the interest of the action. Sometimes, there is no preposition, but I can't explain you why... For a native, it's natural, but it's hard to explain with a rule, with grammar. I wish it's more clear !

Ok it's what I thought This problem is a very serious one, and has not a simple answer. All depends on the verb... one verb can take "à", another "de", or one can take both depending on the context... For exemple the verb "penser": -je pense à elle here we have "à + something" -pense à acheter du pain here we have "à + infinitif" -qu'est ce que tu penses de l'Espagne ? here we have "de + something" -je pense aller à la plage here we have " (-) + infinitif" -je pense que vous devez partir here we have "que + something + infinitif" So for every verbs you should know the construction that fits. It is not random, and it has some logic, but not always. I send you a link that may help you: http://www.connectigramme.com/preposition.html/verbes-prep.html and this one, I think it is the best book for this issue http://www.fichier-pdf.fr/2014/10/04/verbes-et-leurs-prepositions/verbes-et-leurs-prepositions.pdf

This is my final theory so far: * verb + à + infinitif: - denotes objective, purpose, idea of motion towards. * verb + de + infinitif: - denotes specification, coming to an end or stop, idea of motion from. * verb + infinitif: - all the rest

j'espère que la question est plus claire maintenant. Merçi.

Votre question n'est pas assez claire, pouvez-vous préciser quel est votre problème ?