[Matteo]

Hi Marta! Yes sure, here’s an explanation along with some examples to help you grasp these concepts:

Limits: In calculus, limits are used to study the behavior of functions as the input (x-values) approaches a specific value or as it tends to infinity. The limit of a function represents the value that the function approaches as the input gets arbitrarily close to a particular point.

Formally, if we have a function f(x), the limit of f(x) as x approaches a (denoted as lim(x→a) f(x)) is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a.

Continuity: Continuity is a property of a function that indicates the absence of any sudden jumps, breaks, or holes. A function is said to be continuous if it is defined and has no interruptions or gaps over its entire domain.

To determine continuity, we look for three key conditions:

1. The function is defined at the point in question.
2. The limit of the function exists at that point.
3. The value of the function at the point is equal to the limit.

Examples:

1. Limit Example: Consider the function f(x) = 2x + 3. What is the limit of f(x) as x approaches 2?

   • By substituting the value x = 2 into the function, we get f(2) = 2(2) + 3 = 7.
   • So, the limit of f(x) as x approaches 2 is 7 (lim(x→2) f(x) = 7).

2. Continuity Example: Let’s examine the function g(x) = x^2 for all real numbers. Is g(x) continuous?

   • The function g(x) = x^2 is defined for all real numbers.
   • To determine continuity, we need to check if the limit of g(x) exists and if it matches the value of g(x) at each point.
   • Taking the limit as x approaches any real number a, we find that lim(x→a) g(x) = a^2.
   • Since the limit of g(x) exists and matches the value of g(x) at each point, the function g(x) = x^2 is continuous.

If you have any further questions or need additional clarification, feel free to ask. Happy studying!