Let [math]f[/math] be a function defined on an interval around [math]c[/math], although perhaps not at [math]c[/math] itself. The limit of [math]f\left(x\right)[/math] as [math]x[/math] approaches [math]c[/math] is [math]L[/math] if [math]f\left(x\right)[/math] gets really close to [math]L[/math] whenever [math]x[/math] gets really close to [math]c[/math].[br] [br]To say this formally, [math]lim_{x\longrightarrow c}f\left(x\right)=L[/math] if for every [math]\epsilon>0[/math] there exists [math]\delta>0[/math] for which [math]\mid f\left(x\right)-L\mid<\epsilon[/math] whenever [math]0<\mid x-c\mid<\delta[/math].[br][br]The construction below helps make this more intuitive. Pick [math]\epsilon[/math] to set a vertical range for [math]f\left(x\right)[/math] around [math]L[/math]. Then set [math]\delta[/math] small enough that the pink brick fits inside the blue strip.

Let [math]f[/math] be a function defined on an interval around [math]c[/math], although perhaps not at [math]c[/math] itself. The limit of [math]f\left(x\right)[/math] as [math]x[/math] approaches [math]c[/math] is [math]L[/math] if [math]f\left(x\right)[/math] gets really close to [math]L[/math] whenever [math]x[/math] gets really close to [math]c[/math].[br] [br]To say this formally, [math]lim_{x\longrightarrow c}f\left(x\right)=L[/math] if for every [math]\epsilon>0[/math] there exists [math]\delta>0[/math] for which [math]\mid f\left(x\right)-L\mid<\epsilon[/math] whenever [math]0<\mid x-c\mid<\delta[/math].[br][br]The construction below helps make this more intuitive. Pick [math]\epsilon[/math] to set a vertical range for [math]f\left(x\right)[/math] around [math]L[/math]. Then set [math]\delta[/math] small enough that the pink brick fits inside the blue strip.

Let [math]f[/math] be a function defined on an interval around [math]c[/math], although perhaps not at [math]c[/math] itself. The limit of [math]f\left(x\right)[/math] as [math]x[/math] approaches [math]c[/math] is [math]L[/math] if [math]f\left(x\right)[/math] gets really close to [math]L[/math] whenever [math]x[/math] gets really close to [math]c[/math].[br] [br]To say this formally, [math]lim_{x\longrightarrow c}f\left(x\right)=L[/math] if for every [math]\epsilon>0[/math] there exists [math]\delta>0[/math] for which [math]\mid f\left(x\right)-L\mid<\epsilon[/math] whenever [math]0<\mid x-c\mid<\delta[/math].[br][br]The construction below helps make this more intuitive. Pick [math]\epsilon[/math] to set a vertical range for [math]f\left(x\right)[/math] around [math]L[/math]. Then set [math]\delta[/math] small enough that the pink brick fits inside the blue strip.