## Two Special Limits

### There are two special limits in mathematics that are commonly encountered and have unique properties

There are two special limits in mathematics that are commonly encountered and have unique properties. These limits are the limit as x approaches infinity and the limit as x approaches zero. Let’s explore each of them in detail:

1. Limit as x approaches infinity:

The limit as x approaches infinity is denoted as lim(x → ∞) or lim(x → +∞). When we evaluate this limit, we are interested in the behavior of a function f(x) as x gets larger and larger. Here are the possible outcomes when evaluating this limit:

– If the function f(x) grows without bound as x increases, meaning f(x) becomes arbitrarily large as x becomes very large, the limit is said to be divergent or infinite. In this case, we write lim(x → ∞) f(x) = ∞.

– If the function f(x) approaches a specific value L as x becomes infinitely large, we say the limit is convergent. In this case, we write lim(x → ∞) f(x) = L. This means that as x gets larger and larger, f(x) gets arbitrarily close to L.

– In some cases, the function f(x) oscillates or alternates between values as x increases. In these situations, the limit is said to not exist.

2. Limit as x approaches zero:

The limit as x approaches zero is denoted as lim(x → 0) or lim(x → 0+). Here, we are interested in the behavior of a function f(x) as x gets closer and closer to zero. Let’s look at the possible outcomes:

– If the function f(x) approaches a specific value L as x approaches zero, the limit is said to be convergent. In this case, we write lim(x → 0) f(x) = L. This means that as x gets arbitrarily close to zero, f(x) gets arbitrarily close to L.

– If the function f(x) approaches different values as x approaches zero from the left (negative side) and the right (positive side), the limit is said to not exist. In this situation, we say the limit is undefined or divergent.

– Functions can also exhibit oscillatory or alternating behavior as x approaches zero. In these cases, the limit is also said to not exist.

It is important to note that evaluating limits requires careful analysis of the behavior of a function as the independent variable (x) approaches a specific value. By considering the aforementioned outcomes, we can determine if a limit converges or diverges.

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