Understanding the Existence of the Limit as x Approaches a: Exploring Left-Hand and Right-Hand Limits

Show that limit as x approaches a of f(x) exists

To show that the limit as x approaches a of f(x) exists, we need to demonstrate that both the left-hand limit and the right-hand limit of f(x) as x approaches a are equal

To show that the limit as x approaches a of f(x) exists, we need to demonstrate that both the left-hand limit and the right-hand limit of f(x) as x approaches a are equal.

First, let’s consider the left-hand limit. We denote the left-hand limit of f(x) as x approaches a as:

lim(x→a-) f(x)

To show that this limit exists, we need to demonstrate that there is a value L such that as x gets arbitrarily close to a from the left side, f(x) gets arbitrarily close to L.

Formally, for any ε > 0, there exists a δ > 0 such that if 0 < a - x < δ, then |f(x) - L| < ε. Next, let's consider the right-hand limit. We denote the right-hand limit of f(x) as x approaches a as: lim(x→a+) f(x) To show that this limit exists, we need to demonstrate that there is a value M such that as x gets arbitrarily close to a from the right side, f(x) gets arbitrarily close to M. Formally, for any ε > 0, there exists a δ > 0 such that if 0 < x - a < δ, then |f(x) - M| < ε. If we can establish both of these conditions, then we can conclude that the limit as x approaches a of f(x) exists. It's important to note that the values L and M might be equal, in which case the limit is commonly denoted as: lim(x→a) f(x) To prove the existence of the limit, we typically employ techniques such as algebraic manipulation, trigonometric identities, or the definition of limits, depending on the specific function f(x). Overall, the goal is to show that as x approaches a, both the left-hand and right-hand limits of f(x) are equal, indicating the existence of the limit.

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