Limits are a fundamental concept in calculus that allow us to study the behavior of functions as they approach a certain value. They help us answer questions such as 'What is the value of the function at a certain point?' or 'Does the function approach a specific value as the input gets closer to a given point?' To understand limits, let's consider an example.

Consider the function f(x) = (3x^2 + 2x - 1) / (x - 2). As x approaches 2, the denominator (x - 2) approaches 0, which would make the function undefined. However, by evaluating the numerator at x = 2, we find that f(2) = 15. This suggests that the function approaches a value of 15 as x gets arbitrarily close to 2.

In calculus, we use limits to precisely define this behavior. The limit of a function f(x) as x approaches a certain value, typically denoted as 'lim[x → a] f(x)', represents the value that f(x) approaches as x gets arbitrarily close to a. By understanding limits, we can make accurate predictions about the behavior of functions, analyze rates of change, and solve complex problems in various fields of science and engineering.

Remember, limits provide us with powerful tools to understand the behavior of functions and solve intricate problems. Embrace the concept and let's go on a remarkable journey through calculus!