A limit is a the value that a [[Function]] or [[Sequence]] approaches as an input approaches some value. $\huge \lim _{x\to N} f(x) $ >[!example]- >An example: >$ \huge\lim_{x\to6} 1+x= 7 $ >In this case, as x approaches six, $1+x$ approaches 7. >This limit is trivial, and the real use of limits is shown in cases such as this >$ >\huge\lim_{x\to0} \frac{\sin x}{x} >$ >We use a limit to represent this, as $\frac{\sin 0}0 = \text{undefined}.$ However, if we look at a graph of $\frac{\sin x}{x}$, > >Although at $x=0$ the function is undefined, it is clear that as $x$ approaches $0$, the expression gets close, *and stays arbitrarily close* to $1$. That is a mouthful, so we instead write it as >$ >\huge\lim_{x\to0} \frac{\sin x}{x} = 1 >$ > >The way we compute something like this though, would be using [[../../../02 Areas/Math/L'Hôpital's Rule|L'Hôpital's Rule]].