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Geometrically, $\epsilon$ defines a horizontal band around the limit value $L$ on the y-axis, representing the target range for function outputs. $\delta$ defines a vertical interval around the input value $a$ on the x-axis, representing the allowable range for inputs.

Conditions: The graph is plotted on a Cartesian coordinate system.; $\epsilon$ is the half-height of the band around $L$.; $\delta$ is the half-width of the interval around $a$.

Geometrically, $\epsilon$ defines a horizontal band around the limit value $L$ on the y-axis, representing the target range for function outputs. $\delta$ defines a vertical interval around the input value $a$ on the x-axis, representing the allowable range for inputs.

Conditions: The graph is plotted on a Cartesian coordinate system.; $\epsilon$ is the half-height of the band around $L$.; $\delta$ is the half-width of the interval around $a$.

The corner term represents the product of two small changes, $\Delta f \Delta g$. Because the functions are differentiable, both $\Delta f$ and $\Delta g$ are of order $h$ (proportional to the input increment).

Conditions: Functions $f$ and $g$ are differentiable at the point.; The input increment $h$ approaches zero.; The geometric model assumes a rectangle with sides $f(x)$ and $g(x)$.

The corner term represents the product of two small changes, $\Delta f \Delta g$. Because the functions are differentiable, both $\Delta f$ and $\Delta g$ are of order $h$ (proportional to the input increment).

Conditions: Functions $f$ and $g$ are differentiable at the point.; The input increment $h$ approaches zero.; The geometric model assumes a rectangle with sides $f(x)$ and $g(x)$.

The formal definition states that the limit of $f(x)$ as $x$ approaches $a$ is $L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$. This rigorously captures the intuitive idea that $f(x)$ can be made arbitrarily close to $L$ by choosing $x$ sufficiently close to $a$ (but not equal to $a$).

Conditions: $\epsilon$ is an arbitrary positive real number; $\delta$ is a positive real number dependent on $\epsilon$; $x$ is in the domain of $f$ and $x \neq a$

The formal definition states that the limit of $f(x)$ as $x$ approaches $a$ is $L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$. This rigorously captures the intuitive idea that $f(x)$ can be made arbitrarily close to $L$ by choosing $x$ sufficiently close to $a$ (but not equal to $a$).

Conditions: $\epsilon$ is an arbitrary positive real number; $\delta$ is a positive real number dependent on $\epsilon$; $x$ is in the domain of $f$ and $x \neq a$