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 determine the instantaneous rate of change, analyze the existence of limits, and evaluate the behavior of functions at specific points. Let's explore different methods of evaluating limits.

Algebraic Evaluation: One common way to evaluate limits is through algebraic manipulation. For example, consider the limit of the function f(x) = (x^2 - 1)/(x - 1) as x approaches 1. By factoring the numerator and canceling the common factor of (x - 1), we can simplify the function to f(x) = x + 1. Substituting 1 into x, we find that the limit is equal to 2.

Table Evaluation: Another method is to create a table and observe the function's behavior as x approaches a certain value. For instance, let's evaluate the limit of g(x) = sin(x)/x as x approaches 0. By constructing a table with decreasing values of x (e.g., 0.1, 0.01, 0.001, and so on) and calculating the corresponding values of g(x), we notice that the function approaches the value of 1. Therefore, the limit of g(x) as x approaches 0 is equal to 1.

Graphical Evaluation: Graphs provide a visual representation of functions, aiding in limit evaluation. If we visualize the function h(x) = 1/x, we can see that as x approaches 0 from the positive side (x → 0+), the function tends to positive infinity, while as x approaches 0 from the negative side (x → 0-), the function tends to negative infinity. This indicates that the limit of h(x) as x approaches 0 does not exist.

Remember, these are just initial methods to grasp the concept of evaluating limits. As you progress in calculus, you'll learn more advanced techniques like L'Hôpital's rule and the squeeze theorem to evaluate complex limits.

Keep practicing and embracing the beauty of limits!