Understanding the Limit of a Function as x Approaches a Constant: Explained and Conditions for Equality to f(c)

lim x→c f(x) = f(. c)

This statement represents the limit of a function f(x) as x approaches a constant value c

This statement represents the limit of a function f(x) as x approaches a constant value c. The limit of f(x) as x approaches c will be equal to f(c) if certain conditions are met.

To understand this concept further, let’s break it down:

The limit of a function f(x) as x approaches c is denoted as:
lim x→c f(x)

This means that we are interested in finding the behavior of the function f(x) as x gets closer and closer to the constant value c.

The statement “lim x→c f(x) = f(c)” states that the limit of f(x) as x approaches c is equal to the value of f(c).

In simpler terms, if the value of f(x) approaches a specific value f(c) as x gets infinitely close to c, then the limit exists and is equal to f(c).

However, it’s important to note that this statement holds true if and only if the function f(x) is continuous at c.

A function is said to be continuous at c if the following three conditions are satisfied:

1. The function f(x) is defined at c, which means f(c) is a valid value.
2. The limit of f(x) as x approaches c exists.
3. The limit of f(x) as x approaches c is equal to the value of f(c).

If all three conditions are met, then the function f(x) is continuous at c, and we can say that “lim x→c f(x) = f(c)”.

On the other hand, if any of these conditions are not satisfied, then the statement may not hold true, and the limit may not exist, or it may be different from the value of f(c).

In summary, the statement “lim x→c f(x) = f(c)” represents the limit of a function f(x) as x approaches a constant value c, and it states that the limit is equal to the value of f(c) if the function is continuous at c.

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