What is Apparent Authority?
Apparent authority is the power of an agent to act on behalf of his/her principal, based upon the grant of authority that "appears" to an objectively reasonable person.
Example: Owner of hot dog stand pays man to walk down one city block and hold a sign and bark directions to the "best hot dogs in town." A woman hears the man and then asks him how he knows the hot dogs are best. The man says, "Go down and try one and if you don't like it, you don't have to pay."
The woman goes down the street, buys a hot dog, doesn't like it and demands her money back. The vendor refuses.
The woman has a claim against the vendor, because the agent with the sign exhibited the "apparent" authority to offer the money back guarantee, even though the principal/owner, never expressly authorized the agent to make such a guarantee.
The woman also has an independent claim against the agent for a breach of warranty of authority, because he did not have the "actual" authority to offer the guarantee to the woman, but it was objectively reasonable to believe that the agent did have such power.
What is Liability?
A general partnership is an agreement between two or more persons to carry on a business for profit. Absent an express agreement to the contrary, general partners share equal management and control, are entitled to an equal share of the profits of the business, and regardless of any agreement between them, are all jointly liable for the partnership debts.
Limited partners are passive investors who do not have management or control (such control is retained by the general partner(s)), they are only entitled to a specified share of the profits, and they are liable for the partnership debts, only to the extent of their contribution/investment. Limited partnerships are created by statute, must be registered with the state government, and may have different governing rules depending upon the jurisdiction where the partnership is created. General partnerships are a creature of the common law, however, they are also generally regulated by state laws, however they do not require any government registration.
Third parties who deal with general partners, can expect that their dealings will bind the partnership and all of its members, because general partners are agents of the partnership with actual management and control authority. Third parties who deal with limited partners have no such expectation -- only dealing with a general partner can bind the partnership -- except where a limited partner deals as if he/she were actually a general partner, as in acting with management and control over the partnership resources, in which case, the limited partner may exhibit such apparent authority to bind the partnership so as to make it equitable that a third party could reasonably rely on the limited partner's actions, and thereby bind the partnership.
You really ought to do the work yourself. Getting it from someone like myself will cause you to not remember the law, and ultimately, your ability to recall and apply law and legal theory is what will make you a lawyer.