Understanding Infinite Discontinuity in Mathematics: Explained with Examples

Infinite Discontinuity

occurs when a function has a vertical asymptote on one or both sides of an x-value

An infinite discontinuity occurs when the limit of a function as it approaches a point does not exist and the function increases or decreases without bound at that point.

More specifically, suppose we have a function f(x) and a point c. If there exists a limit L such that the limit of f(x) as x approaches c from both the left and right sides is either positive or negative infinity, then we say that f(x) has an infinite discontinuity at c.

The graph of a function with an infinite discontinuity will have a vertical asymptote at the point of discontinuity. This occurs because the function increases or decreases without bound as it approaches the asymptote, but never touches it.

A common example of a function with an infinite discontinuity is the reciprocal function, f(x) = 1/x. This function approaches positive or negative infinity as x approaches 0 from the left or right sides respectively, leading to an infinite discontinuity at x = 0.

It is important to note that an infinite discontinuity is different from a removable discontinuity, which occurs when the limit of a function as it approaches a point exists but the function is not defined at that point.

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