Theorems

The definition of a limit and continuity

A function $y = f(x)$ is continuous at $x = a$, if:

1. $f(a)$ is defined

2. The limit of $f(x)$ as $x$ approaches $a$ exists

   • This implies that both the left-sided and right-sided limits exists and equal the same value: $\lim_{x\to a^{+}}f\left(x\right)=\lim_{x\to a^{-}}f\left(x\right)$ ¹

3. The limit of $f(x)$ as $x$ approaches $a$ is equal to $f(a)$.

Otherwise, $f$ is discontinuous at $x = a$.

¹(The limit exists if both corresponding one sided limits exist and are equal.)

Intermediate-Value Theorem (IVT)

A function $y = f(x)$ that is continuous on a closed interval $[a,b]$ takes on every value between $f(a)$ and $f(b)$.

or

For a function $f$, where $f(x)=y$,

Rolle's Theorem

If a function $f$ is continuous on a closed interval $[a,b]$ and differentiable on $(a,b)$, such that f(a) = f(b), then there is at least one number $c$ in the open interval $(a,b)$ such that $f^{\prime}(c)=0$.

Mean Value Theorem (MVT)

If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is at least one number $c$ in $(a,b)$ such that $\frac{[f(b)-f(a)]}{(b-a)}=f^{\prime}(c)$.

The average rate of change equals the instantaneous rate of change at some point on $(a, b)$.

Extreme Value Theorem

If a function $f$ is continuous on a closed interval $[a,b]$, then $f(x)$ has both a maximum and a minimum on $[a,b]$.

L'Hopital's Rule

If the limit of $\frac{f(x)}{g\left(x\right)}$ as $x$ approaches $a$ is of an indeterminate form, and if the limit of $f'(x) / g'(x)$ as $x$ approaches $a$ exists, then the limit of $f(x) / g(x)$ as $x$ approaches $a$ is equal to the limit of $f'(x) / g'(x)$ as $x$ approaches $a$.