Editing Division by zero (section)

== Calculus ==

[[Calculus]] studies the behavior of [[function (mathematics)|functions]] using the concept of a [[limit (mathematics)|limit]], the value to which a function's output tends as its input tends to some specific value. The notation <math display=inline> \lim_{x \to c} f(x) = L</math> means that the value of the function <math>f</math> can be made arbitrarily close to <math>L</math> by choosing <math>x</math> sufficiently close to <math>c.</math>

In the case where the limit of the [[real function]] <math>f</math> increases without bound as <math>x</math> tends to <math>c,</math> the function is not defined at <math>x,</math> a type of [[mathematical singularity]]. Instead, the function is said to "[[limit (mathematics)#Infinity in limits of functions|tend to infinity]]", denoted <math display=inline> \lim_{x \to c} f(x) = \infty,</math> and its [[Graph of a function|graph]] has the line <math>x=c</math> as a vertical [[asymptote]]. While such a function is not formally defined for <math>x = c,</math> and the [[infinity symbol]] <math>\infty</math> 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", <math>-\infty.</math> In some cases a function tends to two different values when <math>x</math> tends to <math>c</math> from above {{nobr|(<math>x \to c^+</math>)}} and below {{nobr|(<math>x \to c^-</math>)}}; such a function has two distinct [[one-sided limit]]s.<ref>{{citation |last1=Herman |first1=Edwin |chapter-url=https://openstax.org/books/calculus-volume-1/pages/2-2-the-limit-of-a-function |title=Calculus |chapter=2.2 The Limit of a Function |volume=1 |last2=Strang |first2=Gilbert |year=2023 |publisher=OpenStax |isbn=978-1-947172-13-5 |location=Houston |oclc=1022848630 |display-authors=etal |author2-link=Gilbert Strang |page=454}}</ref>

A basic example of an infinite singularity is the [[reciprocal function]], <math>f(x) = 1/x,</math> which tends to positive or negative infinity as <math>x</math> tends to {{nobr|<math>0</math>:}}

<math display=block>
\lim_{x \to 0^+} \frac1x = +\infty,\qquad 
\lim_{x \to 0^-} \frac1x = -\infty.
</math>

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

<math display=block>
\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\displaystyle \lim_{x \to c} f(x)}{\displaystyle \lim_{x \to c} g(x)}.
</math>

However, when a function is constructed by dividing two functions whose separate limits are both equal to <math>0,</math> then the limit of the result cannot be determined from the separate limits, so is said to take an [[indeterminate form]], informally written <math>\tfrac00.</math> (Another indeterminate form, <math>\tfrac \infty \infty,</math> 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 

<math display=block> \lim_{x \to 1} \dfrac{x^2 - 1}{x - 1},</math> 

the separate limits of the numerator and denominator are <math>0</math>, so we have the indeterminate form <math>\tfrac00</math>, but simplifying the quotient first shows that the limit exists: 

<math display=block>
\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.
</math>