If I am a sole member LLC, is my personal assets liable in the case of a lawsuit?

Not unless the party suing you can "pierce the corporate veil" (see below). An LLC is a separate legal person or entity from its owner(s) (its members, to use the correct terms); since it is a separate legal entity, the members are not liable for its debts, obligations, or judgments against it, the same way that person A is not liable for person B's debts or obligations.
(Note: even though you would not be liable simply for being the sole member, you could be sued for your own wrongful acts committed in the course of working for or managing the LLC. Example: if you are driving for business and run someone over, even though you are not liable because it was business travel, you could be liable for having personally been a careless driver.)
"Piercing the corporate veil" means to penetrate the liabilty protection afforded by a corporation or LLC to its shareholders or members. (Since corporations predate LLCs and this doctrine was developed originally for them, the term "piercing the corporate veil" has come to be used for LLCs as well.) Essentially, this means showing that the LLC is a sham--that it does not really have independent existence, but is the "alter ego" of the member(s). If the plaintiff can show this, he can negate the LLC protection. The main things which can be used to pierce the corporate veil is a comingling of funds: depositing payments to the LLC directly into personal accounts; paying personal debts from the LLC account (or vice versa); using LLC property for your own uses; having the LLC take a loss for your personal benefit (like selling you property or assets for below-market rate); etc.--i.e. again, things that show that the LLC is just an extension of you. 
To avoid this, respect the LLC's independent existence: keep its assets, accounts, money, etc. separate. Don't have it bear expenses or take losses for you. If you take money out, due it as a regular salary or as period distributions of profits--it's ok to profit from your LLC, of course (that's why you have one), but you still need to treat the LLC as a separate person that distributes money to you in a predictable, periodic fashion as an employee or member as a consequence of or compensation for your work or ownership, and not as your own "piggy bank."