True or false:

a) if f(x) is continuous at a, then the limit of f(x) as x approaches a exists

b) if the limit of f(x) as x approaches a exists, then f(x) is continuous at a

My answer:
a) false
b) true

To determine the truth value of these statements, let's first understand the definitions of continuity and limits.

a) The statement "if f(x) is continuous at a, then the limit of f(x) as x approaches a exists" is false.

To show this, we need a counterexample. Consider the function f(x) = 1/x when x≠0, and f(x) = 0 when x=0. This function is not continuous at x=0, as the limit of f(x) as x approaches 0 does not exist. However, the function is continuous everywhere else.

b) The statement "if the limit of f(x) as x approaches a exists, then f(x) is continuous at a" is true.

To prove this statement, we can utilize the definition of a limit. The limit of f(x) as x approaches a is defined as the value that f(x) approaches as x gets arbitrarily close to a. If the limit exists, it means that f(x) approaches a well-defined value.

If the limit of f(x) as x approaches a exists, it implies that f(x) is approaching the same value from both sides of a. This corresponds to the definition of continuity, where a function is continuous at a point if the left-hand limit, right-hand limit, and functional value are all equal. Therefore, if the limit of f(x) exists at a point, it guarantees that f(x) is continuous at that point.