Skip to main content
Mathematics LibreTexts

12.6.3: Key Concepts

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Key Concepts

    12.1 Finding Limits: Numerical and Graphical Approaches

    • A function has a limit if the output values approach some value L L as the input values approach some quantity a. a. See Example 1.
    • A shorthand notation is used to describe the limit of a function according to the form lim xa f(x)=L, lim xa f(x)=L, which indicates that as x x approaches a, a, both from the left of x=a x=a and the right of x=a, x=a, the output value gets close to L. L.
    • A function has a left-hand limit if f( x ) f( x ) approaches L L as x x approaches a a where x<a. x<a. A function has a right-hand limit if f( x ) f( x ) approaches L L as x x approaches a a where x>a. x>a.
    • A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
    • A graph provides a visual method of determining the limit of a function.
    • If the function has a limit as x x approaches a, a, the branches of the graph will approach the same y- y- coordinate near x=a x=a from the left and the right. See Example 2.
    • A table can be used to determine if a function has a limit. The table should show input values that approach a a from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See Example 3.
    • A graphing utility can also be used to find a limit. See Example 4.

    12.2 Finding Limits: Properties of Limits

    • The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See Example 1.
    • The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See Example 2 and Example 3.
    • The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See Example 4.
    • The limit of the root of a function equals the corresponding root of the limit of the function.
    • One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See Example 5.
    • Another method of finding the limit of a complex fraction is to find the LCD. See Example 6.
    • A limit containing a function containing a root may be evaluated using a conjugate. See Example 7.
    • The limits of some functions expressed as quotients can be found by factoring. See Example 8.
    • One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See Example 9.

    12.3 Continuity

    • A continuous function can be represented by a graph without holes or breaks.
    • A function whose graph has holes is a discontinuous function.
    • A function is continuous at a particular number if three conditions are met:
      • Condition 1: f(a) f(a) exists.
      • Condition 2: lim xa f(x) lim xa f(x) exists at x=a. x=a.
      • Condition 3: lim xa f(x)=f(a). lim xa f(x)=f(a).
    • A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
    • A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See Example 1.
    • Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See Example 2 and Example 3.
    • For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See Example 4 and Example 5.

    12.4 Derivatives

    • The slope of the secant line connecting two points is the average rate of change of the function between those points. See Example 1.
    • The derivative, or instantaneous rate of change, is a measure of the slope of the curve of a function at a given point, or the slope of the line tangent to the curve at that point. See Example 2, Example 3, and Example 4.
    • The difference quotient is the quotient in the formula for the instantaneous rate of change:
      f( a+h )f( a ) h f( a+h )f( a ) h
    • Instantaneous rates of change can be used to find solutions to many real-world problems. See Example 5.
    • The instantaneous rate of change can be found by observing the slope of a function at a point on a graph by drawing a line tangent to the function at that point. See Example 6.
    • Instantaneous rates of change can be interpreted to describe real-world situations. See Example 7 and Example 8.
    • Some functions are not differentiable at a point or points. See Example 9.
    • The point-slope form of a line can be used to find the equation of a line tangent to the curve of a function. See Example 10.
    • Velocity is a change in position relative to time. Instantaneous velocity describes the velocity of an object at a given instant. Average velocity describes the velocity maintained over an interval of time.
    • Using the derivative makes it possible to calculate instantaneous velocity even though there is no elapsed time. See Example 11.

    12.6.3: Key Concepts is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?