Limits are an important idea in calculus. They help us understand what happens to a function when we get closer to a certain number. When we talk about infinity, limits are used to see how functions behave when they go towards really big or really small values. Here are some key points about limits and infinity:

1. Understanding Asymptotic Behavior:

   • Limits help us find horizontal or vertical lines that graphs approach but never actually touch. For example, if we look at the function ( f(x) = \frac{1}{x} ) as ( x ) gets really big, the limit is ( 0 ). This means the graph gets close to the x-axis but never reaches it.

2. Evaluating Infinite Limits:

   • When we check the function ( f(x) = x^2 ) as ( x ) goes to infinity, we see that ( \lim_{x \to \infty} f(x) = \infty ). This means the function just keeps getting bigger and bigger without stopping.

3. Real-World Applications:

   • Physics: Limits help us figure out how things change at a specific moment, like finding the speed of an object. We can do this by looking at the limit of average speed as the time period gets very small.
   • Economics: In money matters, limits can explain costs and profits as production gets close to a maximum level.
   • Biology: In studying populations, limits help us understand how many individuals a certain environment can support, showing how growth approaches a maximum population size.

4. Continuous Functions:

   • Limits are crucial for understanding continuous functions. A function is continuous at a point if the limit as ( x ) gets closer to that point is the same as the function's value at that point. This is very important when we apply math to real life, ensuring that our models react smoothly to changes.

Learning about limits not only makes math easier but also prepares students to use these ideas in many areas, connecting math theory with real-life situations.