Answer: If the limit of a function f(x) exists as x approaches a, then both the left-hand limit (limx→a- f(x)) and the right-hand limit (limx→a+ f(x)) also exist.
Explanation:
1. To understand why this is true, let's start with the definition of a limit. The limit of f(x) as x approaches a, denoted as limx→a f(x), represents the value that f(x) approaches as x gets arbitrarily close to a.
2. The left-hand limit (limx→a- f(x)) represents the value that f(x) approaches as x approaches a from the left side, or values of x that are smaller than a. This is the limit as x gets arbitrarily close to a but from values smaller than a.
3. Similarly, the right-hand limit (limx→a+ f(x)) represents the value that f(x) approaches as x approaches a from the right side, or values of x that are larger than a. This is the limit as x gets arbitrarily close to a but from values larger than a.
4. Since the limit of f(x) exists as x approaches a, it means that f(x) approaches a unique value as x gets arbitrarily close to a. This implies that both the left-hand limit and the right-hand limit must also approach the same unique value.
5. Therefore, if limx→a f(x) exists, then both limx→a- f(x) and limx→a+ f(x) also exist and have the same value.
In summary, if the limit of a function f(x) exists as x approaches a, then the left-hand limit and the right-hand limit of f(x) also exist and have the same value.
I hope this helps :)