There are 4 definitions of limits. The first one is the definition of finite limit. This definition is important when we want to understand the behavior of a function as it approaches a specific value.
The Finite Limit Definition
In the finite limit definition, we are concerned with the behavior of a function f(x) as x tends to a specific value, denoted as xo. The function f(x) has a finite limit as x approaches xo if and only if for any arbitrarily small and positive E, there exists a neighborhood of xo such that for every x in that neighborhood (except xo), f(x) falls within a circular neighborhood of E centered at P.
For example, if we take the function f(x) = x²-3x+2, and we want to find the limit of f(x) as x tends to 2 (i.e., xo=2), we can do so by first showing that for a suitable E, and for x close to 2 but not equal to 2, the function f(x) falls within a given range. If this condition is satisfied, then the limit exists.
The Limits Verification Process
The verification of a limit involves three fundamental steps. We first fix a suitable E value, then we examine where the function falls within the given neighborhood, and finally we check if the function satisfies the conditions of the limit definition within the neighborhood.
The Uniqueness of Limits Theorem
The uniqueness of limits theorem states that if the limit of a function f(x) exists as x tends to xo, then this limit is unique. This means that there cannot be more than one limit for a function approaching a specific value.
The theorem can be demonstrated by assuming the contrary. We suppose that there are two different limits, l and l', for the same function as x approaches xo. By using the definition of the finite limit and by carefully selecting suitable values for E, it can be shown that the two limits must be equal, which contradicts the initial assumption.
The Comparison Theorem
The comparison theorem, also known as the "two policemen theorem", is used to demonstrate the uniqueness of limits. It shows that if the limits of two functions g(x) and h(x) exist as x approaches xo, and if g(x) is always less than or equal to h(x) for x in the neighborhood of xo, then the limit of g(x) must be less than or equal to the limit of h(x).
By carefully selecting suitable values for E and using the definition of finite limits, we can verify that the limit of g(x) must indeed be less than or equal to the limit of h(x), which supports the uniqueness of limits theorem.
In conclusion, the study of limits is crucial in understanding the behavior of a function as it approaches a specific value. The theorems and definitions presented here help ensure that the limit of a function is well-defined and unique, providing a solid foundation for further mathematical analysis.
