Question

Determine whether the statement is true or false, and explain why.

The limit of f(x) as x-a describes what
happens to f(x) when x is near, but not equal to, the value a.
Choose the correct answer below.
limit of x as f(x)→a OC. This statement is false because the limit
A. This statement is false because the true because it is part of the of f(x) as x-a describes what happens to f(x) when x is equal to a.
B. This statement is OD. This statement is false because the
describes what happens to x when f(x) is equal to a. limit of f(x) as x-a does not exist
definition of the limit. for all f(x) and all a.

Answer 1

The statement about the limit of f(x) as x approaches a being related to the values of f(x) near, but not equal to a is true, reflecting the behavior of a function near a specific point without requiring the function to actually reach that value.

Step-by-step explanation:

This is because the definition of a limit in mathematics is specifically concerned with the behavior of a function as the input value approaches, but does not necessarily reach a certain point. For instance, if we have a horizontal line represented by f(x) = 20 for a domain of 0 ≤ x ≤ 20, the limit of f(x) as x approaches any value a within that domain is indeed 20, regardless of whether x actually equals a or is merely approaching it. It is important to understand that a limit does not give information about the function's value at the point x = itself, which might or might not equal the limit. Additionally, not all limits exist, such as when a function approaches infinity or behaves erratically near the point of interest. Furthermore, contexts like physics may use the concept of limits to describe situations where quantities approach but do not reach certain values, such as acceleration approaching zero as the difference in velocities diminishes.