In AD&D 1e, the number of weapons you know how to use is given in the weapon proficiency table on page 37 of the Players Handbook. As a first-level magic-user, you can know how to use one weapon. Other weapons that you can also use, you use at the “non-proficiency penalty”, which in your case is a penalty of 5.
At level seven, you can know how to use 2 weapons without the penalty; at level thirteen, three (see Added Proficiency on page 37). You can still use any weapon your DM allows you to use, with the penalty.
Look on page 19 of the Players Handbook for the list of weapons your magic-user is “permitted” to know how to use: dagger, dart, and staff. Unearthed Arcana expands this list to “caltrop, dagger, dart, knife, sling, staff” (see page 13).
AD&D 1e is very different from 3.5 and later editions, philosophically. Except for certain specific instances such as clerics and bladed weapons, I don’t think there is any place in the core books that say “you cannot use any weapon not on the list” or “you can use any weapon, but if they’re not on the list you can’t know how to use them, so you’ll always get the penalty.”
Most of the games I’ve played in, if the matter came up (it rarely did) used the latter interpretation, which is bolstered by the note on page 36 of the Players Handbook that “If proficiency with any given weapon is not held by the character, it is used at a penalty as shown on the table which follows.” But that’s thin support, and neither the text nor general practice mean that a particular ruling is correct; there is a lot of interpretation in AD&D 1e.
Stronger support for both positions is in Unearthed Arcana (page 13), that “Weapons not permitted to a character of a certain class may be picked up and used, but the non-proficiency penalty for that class will always apply (using the most severe penalty applicable for multi-classed characters)—and such non-permitted weapons may not be kept and carried by the character for use later.” Under this interpretation, your magic-user can use, say, a crossbow that happens to be in their vicinity, but cannot actually carry a crossbow so that one is always available.
That said, the list of permitted weapons tends to be very fluid per campaign. I have, written in pencil on page 19 of my Players Handbook, “sai” listed after “dagger, dart, and staff”. And Oriental Adventures has a very different list on page 14. It is completely up to the DM whether the permitted list is used verbatim or whether it is added to or even changed, often (as in Oriental Adventures) in accordance with genre or culture.
It sounds like your DM has changed it—sling isn’t in the default list for AD&D 1e, unless there were different printings I’m unaware of (depending on which weapons you don’t remember, it may also be that your DM is using the Unearthed Arcana list). This was perfectly normal. Almost everything in AD&D 1e except the name of the game and the author of the books was considered open to interpretation.
So if you think your magic-user ought to be able to use some other weapon, bring it up with the GM. However, ranged weapons are more likely to be refused than simple melee weapons, because as a magic-user you are expected to use magic—or weird ideas—at range.
As a player, you are neither required nor expected to look in the Dungeon Masters Guide. The information in the Guide is expected to be filtered through your DM to you. “What follows herein is strictly for the eyes of you, the campaign referee… As this book is the exclusive precinct of the DM, you must view any non-DM player possessing it as something less than worthy of honorable death… It is in your interests, and in theirs, to discourage possession of this book by players.” (page 7-8 of the Guide)
Which doesn’t mean that players don’t often read it, but except for a few specific rules such as saving throws and combat charts, the rules are meant to be even more open to interpretation than the Players Handbook. Looking in the Guide if the Handbook is confusing is only likely to be even more confusing. I could not find anything in the Guide that speaks specifically to your question.