Limits and continuity are fundamental concepts in calculus that help us analyze the behavior of functions. Let's start by understanding what a limit is.

A limit is the value a function approaches as the input approaches a certain value. We denote a limit by using the notation lim, followed by the function and the value towards which the input is approaching. For example, lim(x->2) f(x) represents the limit of the function f(x) as x approaches 2.

To calculate limits algebraically, we substitute the approaching value into the function and simplify. Let's consider the function f(x) = 2x + 3. To find lim(x->1) f(x), we substitute x = 1 into the function: f(1) = 2(1) + 3 = 5. Hence, lim(x->1) f(x) = 5.

Continuity is closely related to limits. A function is continuous if it has no breaks, holes, or jumps in its graph. In other words, as we move along the graph, there are no sudden changes or discontinuities. A function can be continuous at a specific point, on an interval, or over its entire domain.

Remember, limits and continuity provide us with powerful tools to analyze functions and understand their behavior. Keep practicing and exploring more examples, and you'll master this concept in no time!