In mathematics, limits and continuity play a crucial role in understanding the behavior of functions. They help us analyze how functions behave as their inputs approach certain values, and whether there are any breaks or jumps in their graphs. Let's dive into the world of limits and continuity with some examples.

Example 1: Limit of a Function

Consider the function f(x) = 2x + 3. We want to find the limit of f(x) as x approaches 2. We substitute x = 2 into the function and evaluate it: f(2) = 2(2) + 3 = 7. So, the limit of f(x) as x approaches 2 is 7.

Example 2: Discontinuous Function

Now, let's examine a discontinuous function. Consider the function g(x) = 1/x. As x approaches 0, the function becomes undefined because division by zero is not possible. Therefore, g(x) is discontinuous at x = 0.

Example 3: Continuity of a Function

A function is said to be continuous if there are no jumps, breaks, or holes in its graph. For example, the function h(x) = x^2 is continuous for all real numbers. There are no breaks or jumps in its graph, and it is smooth and connected.

Remember, limits help us determine the behavior of functions, and continuity ensures the smoothness of a function's graph. With practice, you'll master these concepts and apply them to various mathematical problems.

So, keep exploring the exciting world of limits and continuity in mathematics. Remember, math is fun!