Understanding Limits of Functions: Analyzing Behavior as Values Approach a Specific Point

lim x→c f(x) exists

When discussing the limit of a function as it approaches a certain value, it means analyzing the behavior of the function as the input values get arbitrarily close to that specific value

When discussing the limit of a function as it approaches a certain value, it means analyzing the behavior of the function as the input values get arbitrarily close to that specific value.

In this case, “lim x→c f(x) exists” means that the limit of the function f(x) exists as x approaches the value c, being c a constant.

To determine if the limit exists, we need to check two conditions: the left-hand limit and the right-hand limit.

1. Left-hand limit: We analyze the behavior of the function as x approaches c from the left side (x < c). To find the left-hand limit, we write it as: lim x→c^- f(x). 2. Right-hand limit: We analyze the behavior of the function as x approaches c from the right side (x > c).
To find the right-hand limit, we write it as: lim x→c^+ f(x).

If both the left-hand limit and the right-hand limit are equal, then the limit of the function as x approaches c exists. Symbolically, if lim x→c^- f(x) = lim x→c^+ f(x), then lim x→c f(x) exists.

It’s important to note that if the left-hand limit and right-hand limit do not exist or are not equal, then the overall limit does not exist.

To further analyze the limit, additional methods such as substitution, factoring, or using limit rules may be necessary depending on the specifics of the function f(x).

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