Mathematical Existence: Prohibitions On Division By Zero And Imaginary Numbers

Two ways a function can’t exist

If a limit of a function is an unstable number and it keeps changing, such as sin (π/x) which forever oscillates between -1 and 1 and x gets closer to 0, then the limit doesn’t exist. Also, if the number the limit is equal to gets insanely large, such as 1/x^2. Also, obviously x can’t be zero in either of these because you can’t divide by zero.

1. Division by zero: If a function involves division by zero, it cannot exist. Dividing any number by 0 is undefined and is not allowed in mathematics. For example, consider the function f(x) = 1/x. This function cannot exist at x = 0 because division by zero is not allowed.

2. Imaginary or complex numbers: If a function gives imaginary or complex numbers as output, it cannot exist for real-valued inputs. For example, the function f(x) = √(x+2) gives a complex number as output for x < -2. Since we usually work with real numbers, a function that gives complex numbers as outputs cannot exist for real-valued inputs. Note: Sometimes, functions that cannot exist for some inputs can be extended to include those inputs in other contexts. For example, the function f(x) = 1/x can be extended to include x=0 by defining f(0) = ∞, resulting in what is called a removable discontinuity. However, this extension is not always mathematically meaningful or useful.

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