Welcome to the fascinating world of limits and continuity in calculus! These two fundamental concepts play a crucial role in understanding the behavior of functions and solving complex mathematical problems. Limits provide a way to describe what happens to a function as it approaches a certain point, while continuity deals with the smoothness and connectedness of a function's graph.

To better grasp limits, consider the function f(x) = (x^2 - 1) / (x - 1). As x approaches 1, the denominator approaches 0, which would typically result in division by zero. However, by factoring the numerator and canceling out common factors, we find that f(x) simplifies to x + 1. This expression then represents the limit of f(x) as x approaches 1.

Limits are evaluated using various techniques, including direct substitution, factoring, rationalizing, and applying trigonometric identities, among others. Each method has its strengths depending on the complexity of the function being analyzed. As we progress, we will explore these approaches in greater detail, enabling you to confidently evaluate limits with ease.

Remember, perseverance and practice are the keys to mastering limits and continuity. Don't be discouraged if certain concepts seem challenging at first. With determination and effort, you will soon appreciate the elegance and power of these mathematical principles. Keep up the great work!