Precalculus

Key Concepts

PrecalculusKey Concepts

12.1Finding 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 $lim x→a f(x)=L, lim x→a f(x)=L,$ 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 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.

12.2Finding 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.

12.3Continuity

• 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: $lim x→a f(x) lim x→a f(x)$ exists at $x=a. x=a.$
• Condition 3: $lim x→a f(x)=f(a). lim x→a f(x)=f(a).$
• 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.

12.4Derivatives

• 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.
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