### Key Terms

- average velocity
- the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval $\left[t,a\right]$ (if $t<a$ or $\left[a,t\right]$ if $t>a)$, with a position given by $s\left(t\right),$ that is ${v}_{\text{ave}}=\frac{s\left(t\right)-s\left(a\right)}{t-a}$

- constant multiple law for limits
- the limit law $\underset{x\to a}{\text{lim}}cf\left(x\right)=c\xb7\underset{x\to a}{\text{lim}}f\left(x\right)=cL$

- continuity at a point
- A function $f\left(x\right)$ is continuous at a point
*a*if and only if the following three conditions are satisfied: (1) $f\left(a\right)$ is defined, (2) $\underset{x\to a}{\text{lim}}f\left(x\right)$ exists, and (3) $\underset{x\to a}{\text{lim}}f\left(x\right)=f\left(a\right)$

- continuity from the left
- A function is continuous from the left at
*b*if $\underset{x\to {b}^{-}}{\text{lim}}f\left(x\right)=f\left(b\right)$

- continuity from the right
- A function is continuous from the right at
*a*if $\underset{x\to {a}^{+}}{\text{lim}}f(x)=f(a)$

- continuity over an interval
- a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function $f\left(x\right)$ is continuous over a closed interval of the form $\left[a,b\right]$ if it is continuous at every point in $\left(a,b\right),$ and it is continuous from the right at
*a*and from the left at*b*

- difference law for limits
- the limit law $\underset{x\to a}{\text{lim}}\left(f\left(x\right)-g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)-\underset{x\to a}{\text{lim}}g\left(x\right)=L-M$

- differential calculus
- the field of calculus concerned with the study of derivatives and their applications

- discontinuity at a point
- A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point

- epsilon-delta definition of the limit
- $\underset{x\to a}{\text{lim}}f\left(x\right)=L$ if for every $\epsilon >0,$ there exists a $\delta >0$ such that if $0<\left|x-a\right|<\delta ,$ then $\left|f\left(x\right)-L\right|<\epsilon $

- infinite discontinuity
- An infinite discontinuity occurs at a point
*a*if $\underset{x\to {a}^{-}}{\text{lim}}f\left(x\right)=\text{\xb1}\infty $ or $\underset{x\to {a}^{+}}{\text{lim}}f\left(x\right)=\text{\xb1}\infty $

- infinite limit
- A function has an infinite limit at a point
*a*if it either increases or decreases without bound as it approaches*a*

- instantaneous velocity
- The instantaneous velocity of an object with a position function that is given by $s\left(t\right)$ is the value that the average velocities on intervals of the form $\left[t,a\right]$ and $\left[a,t\right]$ approach as the values of
*t*move closer to $a,$ provided such a value exists

- integral calculus
- the study of integrals and their applications

- Intermediate Value Theorem
- Let
*f*be continuous over a closed bounded interval $\left[\text{a},\text{b}\right];$ if*z*is any real number between $f\left(a\right)$ and $f\left(b\right),$ then there is a number*c*in $\left[a,b\right]$ satisfying $f\left(c\right)=z$

- intuitive definition of the limit
- If all values of the function $f\left(x\right)$ approach the real number
*L*as the values of $x\left(\ne a\right)$ approach*a*, $f\left(x\right)$ approaches*L*

- jump discontinuity
- A jump discontinuity occurs at a point
*a*if $\underset{x\to {a}^{-}}{\text{lim}}f\left(x\right)$ and $\underset{x\to {a}^{+}}{\text{lim}}f\left(x\right)$ both exist, but $\underset{x\to {a}^{-}}{\text{lim}}f\left(x\right)\ne \underset{x\to {a}^{+}}{\text{lim}}f\left(x\right)$

- limit
- the process of letting
*x*or*t*approach*a*in an expression; the limit of a function $f\left(x\right)$ as*x*approaches*a*is the value that $f\left(x\right)$ approaches as*x*approaches*a*

- limit laws
- the individual properties of limits; for each of the individual laws, let $f\left(x\right)$ and $g\left(x\right)$ be defined for all $x\ne a$ over some open interval containing
*a*; assume that*L*and*M*are real numbers so that $\underset{x\to a}{\text{lim}}f\left(x\right)=L$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=M;$ let*c*be a constant

- multivariable calculus
- the study of the calculus of functions of two or more variables

- one-sided limit
- A one-sided limit of a function is a limit taken from either the left or the right

- power law for limits
- the limit law $\underset{x\to a}{\text{lim}}{\left(f\left(x\right)\right)}^{n}={\left(\underset{x\to a}{\text{lim}}f\left(x\right)\right)}^{n}={L}^{n}$ for every positive integer
*n*

- product law for limits
- the limit law $\underset{x\to a}{\text{lim}}\left(f\left(x\right)\xb7g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)\xb7\underset{x\to a}{\text{lim}}g\left(x\right)=L\xb7M$

- quotient law for limits
- the limit law $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to a}{\text{lim}}f\left(x\right)}{\underset{x\to a}{\text{lim}}g\left(x\right)}=\frac{L}{M}$ for $M\ne 0$

- removable discontinuity
- A removable discontinuity occurs at a point
*a*if $f\left(x\right)$ is discontinuous at*a*, but $\underset{x\to a}{\text{lim}}f\left(x\right)$ exists

- root law for limits
- the limit law $\underset{x\to a}{\text{lim}}\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to a}{\text{lim}}f\left(x\right)}=\sqrt[n]{L}$ for all
*L*if*n*is odd and for $L\ge 0$ if*n*is even

- secant
- A secant line to a function $f\left(x\right)$ at
*a*is a line through the point $\left(a,f\left(a\right)\right)$ and another point on the function; the slope of the secant line is given by ${m}_{\text{sec}}=\frac{f\left(x\right)-f\left(a\right)}{x-a}$

- squeeze theorem
- states that if $f\left(x\right)\le g\left(x\right)\le h\left(x\right)$ for all $x\ne a$ over an open interval containing
*a*and $\underset{x\to a}{\text{lim}}f\left(x\right)=L=\underset{x\to a}{\text{lim}}h\left(x\right)$ where*L*is a real number, then $\underset{x\to a}{\text{lim}}g\left(x\right)=L$

- sum law for limits
- The limit law $\underset{x\to a}{\text{lim}}\left(f\left(x\right)+g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)+\underset{x\to a}{\text{lim}}g\left(x\right)=L+M$

- tangent
- A tangent line to the graph of a function at a point $\left(a,f\left(a\right)\right)$ is the line that secant lines through $\left(a,f\left(a\right)\right)$ approach as they are taken through points on the function with
*x*-values that approach*a*; the slope of the tangent line to a graph at*a*measures the rate of change of the function at*a*

- triangle inequality
- If
*a*and*b*are any real numbers, then $\left|a+b\right|\le \left|a\right|+\left|b\right|$

- vertical asymptote
- A function has a vertical asymptote at $x=a$ if the limit as
*x*approaches*a*from the right or left is infinite