Division by zero

Calculus[edit]

Calculus studies the behavior of functions using the concept of a limit, the value to which a function's output tends as its input tends to some specific value. The notation ${\textstyle \lim _{x\to c}f(x)=L}$ means that the value of the function ${\displaystyle f}$ can be made arbitrarily close to ${\displaystyle L}$ by choosing ${\displaystyle x}$ sufficiently close to ${\displaystyle c.}$


In the case where the limit of the real function ${\displaystyle f}$ increases without bound as ${\displaystyle x}$ tends to ${\displaystyle c,}$ the function is not defined at ${\displaystyle x,}$ a type of mathematical singularity. Instead, the function is said to "tend to infinity", denoted ${\textstyle \lim _{x\to c}f(x)=\infty ,}$ and its graph has the line ${\displaystyle x=c}$ as a vertical asymptote. While such a function is not formally defined for ${\displaystyle x=c,}$ and the infinity symbol ${\displaystyle \infty }$ in this case does not represent any specific real number, such limits are informally said to "equal infinity". If the value of the function decreases without bound, the function is said to "tend to negative infinity", ${\displaystyle -\infty .}$ In some cases a function tends to two different values when ${\displaystyle x}$ tends to ${\displaystyle c}$ from above (${\displaystyle x\to c^{+}}$) and below (${\displaystyle x\to c^{-}}$); such a function has two distinct one-sided limits.^[23]


A basic example of an infinite singularity is the reciprocal function, ${\displaystyle f(x)=1/x,}$ which tends to positive or negative infinity as ${\displaystyle x}$ tends to ${\displaystyle 0}$:


$${\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x}}=+\infty ,\qquad \lim _{x\to 0^{-}}{\frac {1}{x}}=-\infty .}$$


In most cases, the limit of a quotient of functions is equal to the quotient of the limits of each function separately,


$${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {\displaystyle \lim _{x\to c}f(x)}{\displaystyle \lim _{x\to c}g(x)}}.}$$


However, when a function is constructed by dividing two functions whose separate limits are both equal to ${\displaystyle 0,}$ then the limit of the result cannot be determined from the separate limits, so is said to take an indeterminate form, informally written ${\displaystyle {\tfrac {0}{0}}.}$ (Another indeterminate form, ${\displaystyle {\tfrac {\infty }{\infty }},}$ results from dividing two functions whose limits both tend to infinity.) Such a limit may equal any real value, may tend to infinity, or may not converge at all, depending on the particular functions. For example, in


$${\displaystyle \lim _{x\to 1}{\dfrac {x^{2}-1}{x-1}},}$$


the separate limits of the numerator and denominator are ${\displaystyle 0}$, so we have the indeterminate form ${\displaystyle {\tfrac {0}{0}}}$, but simplifying the quotient first shows that the limit exists:


$${\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=\lim _{x\to 1}{\frac {(x-1)(x+1)}{x-1}}=\lim _{x\to 1}(x+1)=2.}$$