Do Functions with the Same Limit at xc Necessarily Mean They are Equal?

Do Functions with the Same Limit at xc Necessarily Mean They are Equal?

At first glance, one might be tempted to think that if two functions share the same limit at a specific point, then they must be the same function. However, this is not necessarily the case. Functions can share the same limit at a point without being identical. In this article, we will explore why and provide practical examples to illustrate this concept.

Introduction and Definitions

Let us start with the fundamental definitions:

Limits: The limit of a function as ( x ) approaches ( c ) is a fixed number ( L ) such that as ( x ) gets arbitrarily close to ( c ), the function's value approaches ( L ). Equal Functions: Two functions are considered equal if and only if they have the same domain, the same codomain, and they yield the same value for every ( x ) in the domain. Continuity: A function is continuous at a point if it is well-behaved at that point (no breaks, jumps, or holes).

Example with Trigonometric and Linear Functions

Consider the functions ( f_1(x) sin x ) and ( f_2(x) x ). Both functions share a limit of 0 as ( x ) approaches 0:

Example: [lim_{xto 0} sin x 0 lim_{xto 0} x ]

However, it is clear that these functions are not equal for all ( x ), as demonstrated by evaluating at ( x pi ), for instance:

[f_1(pi) sin(pi) 0 quad text{and} quad f_2(pi) pi eq 0 ]

This example illustrates a key point: two functions can have the same limit at a particular point but not be the same function overall.

Example of Functions with the Same Limit but Different Behavior

Let us provide a more concrete example to solidify this concept:

Consider the following two functions:

[f_1(x) 4x^2 quad text{and} quad f_2(x) 1 4x]

Both functions are continuous for all ( x in [-alpha, alpha] ).

At ( x 3 ), we have:

[f_1(3) 4(3^2) 36 quad text{and} quad f_2(3) 1 4(3) 13]

Note that even though ( f_1(3) f_2(3) ), the functions clearly differ in behavior elsewhere. To determine if two functions are equal, we need to check if they are identical throughout their domain, as follows:

[f_1(x) 4x^2 quad text{and} quad f_2(x) 1 4x quad text{for all} quad x in [-alpha, alpha]]

Since ( 4x^2 eq 1 4x ) for most values of ( x ), these functions are not equal. The key takeaway here is that even if two functions have the same value at a certain point, they are not necessarily equal everywhere in their domain.

Example of Discontinuous Functions with the Same Limit

Let us also consider a pair of piecewise functions:

[f(x) begin{cases} 1 text{if } x 0 x text{otherwise} end{cases} quad text{and} quad g(x) begin{cases} -1 text{if } x 0 -x text{otherwise} end{cases}]

Both ( f(x) ) and ( g(x) ) have a limit of 0 as ( x ) approaches 0:

[lim_{xto 0} f(x) 0 quad text{and} quad lim_{xto 0} g(x) 0]

However, due to their piecewise nature, ( f(x) eq g(x) ) for all ( x ). Specifically, ( f(0) 1 ) and ( g(0) -1 ), and ( f(x) eq g(x) ) for all other ( x ).

Therefore, even functions with the same limit at a point can differ significantly in their behavior and cannot be considered equal functions.

Conclusion

In summary, two functions can have the same limit at a specific point without being identical. The equality of functions depends on far more than just having the same limit at a single point. The concept of domain equality, codomain equality, and identical behavior across the entire domain are crucial for determining if two functions are equal.

Understanding these concepts is essential in mathematics, especially in fields like calculus, where functions and their properties play a critical role.