I generally tell my students that, in laymen terms, to say lim as x-->c of f(x) = L just means that when x is close to c, BUT NOT EQUAL TO C, then f(x) is close to L. This is easy to view graphically, and matches up with any function that is continuous at x = c, or has a removable discontinuity at x = c.
Thus, the limit is just a y-value (for functions of one variable) that your function approaches, though the function may or may not take on the value of L when x = c.
To fully understand limits of functions of one variable, review the epsilon-delta definition carefully.
Though the definition of limits expands in a natural way to multi-variable functions, evaluating limits for functions of 2 or more variables can get much more complicated. Understanding the relationships between the geometric and algebraic properties of limits also plays a huge role, as with many mathematical topics.
It is good to delve into the topic of limits. It is the foundation of all calculus. The definitions of the derivative, and of the anti-derivative, of a function are both limits, and being able to apply calculus to physics requires an understanding of the underlying geometry of limits.