When Does a Limit Not Exist: Exploring Cases of Non-Existent Limits in Mathematics

when does a limit not exist

A limit does not exist in several cases:

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A limit does not exist in several cases:

1. The function approaches different values from the left and right sides of the point. In other words, the left-hand limit and the right-hand limit at a particular point do not match. For example, consider the function f(x) = 1/x. The limit of f(x) as x approaches 0 from the left (x → 0-) is negative infinity, while the limit as x approaches 0 from the right (x → 0+) is positive infinity. Since these limits do not match, the limit of f(x) as x approaches 0 does not exist.

2. The function is oscillating or discontinuous at the point of interest. If the function constantly switches between different values as it gets closer to the point, the limit does not exist. For example, consider the function g(x) = sin(1/x). As x approaches 0, the value of g(x) oscillates between -1 and 1, never settling on a specific value. Therefore, the limit of g(x) as x approaches 0 does not exist.

3. The function approaches infinity or negative infinity as x approaches a certain value. If the function grows without bound or approaches negative infinity as x approaches a particular value, the limit is said to be infinite or non-existent. For example, consider the function h(x) = 1/x^2. As x approaches 0, the value of h(x) approaches positive infinity. Therefore, the limit of h(x) as x approaches 0 is infinite and does not exist.

These are just a few examples of situations where a limit does not exist. It’s important to carefully analyze the behavior of the function near the point of interest to determine whether a limit exists or not.

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