Understanding the Limit of a Function as x Approaches c

lim x→c f(x) exists

When we say that the limit of a function f(x) as x approaches c exists, it means that as x gets arbitrarily close to c, the values of f(x) approach a certain number or “limit”

When we say that the limit of a function f(x) as x approaches c exists, it means that as x gets arbitrarily close to c, the values of f(x) approach a certain number or “limit”.

To formally define the limit of f(x) as x approaches c, we need to consider the behavior of f(x) as x gets closer and closer to c from both the left and the right sides.

First, let’s define the left-hand limit of f(x) as x approaches c, denoted as:

lim x→c- f(x)

This represents the behavior of f(x) as x approaches c from values less than c. To determine the left-hand limit, we evaluate f(x) for values of x that are smaller than c. If these function values approach a certain number as x gets closer to c, then the left-hand limit exists.

Similarly, let’s define the right-hand limit of f(x) as x approaches c, denoted as:

lim x→c+ f(x)

This represents the behavior of f(x) as x approaches c from values greater than c. To determine the right-hand limit, we evaluate f(x) for values of x that are greater than c. If these function values approach a certain number as x gets closer to c, then the right-hand limit exists.

Now, if both the left-hand limit and the right-hand limit exist and are equal, denoted as:

lim x→c- f(x) = lim x→c+ f(x)

then we can say the limit of f(x) as x approaches c exists and is equal to this common value.

In terms of notation, we express the limit of f(x) as x approaches c as:

lim x→c f(x)

where the limit is taken from both sides (approaching c from values less than c and values greater than c).

It’s important to note that the limit of f(x) as x approaches c does not depend on the actual value of f(c), but rather on the behavior of f(x) as x approaches c. f(c) may or may not be equal to the limit, as they represent different concepts.

Overall, if the limit of f(x) as x approaches c exists, it means that the function has a well-defined value or tendency as x gets arbitrarily close to c.

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