12.1 Finding Limits: Numerical and Graphical Approaches

A function has a limit if the output values approach some value $L$ as the input values approach some quantity $a .$ See Example 1.
A shorthand notation is used to describe the limit of a function according to the form $\lim_{x \to a} f (x) = L,$ which indicates that as $x$ approaches $a,$ both from the left of $x = a$ and the right of $x = a,$ the output value gets close to $L .$
A function has a left-hand limit if $f (x)$ approaches $L$ as $x$ approaches $a$ where $x < a .$ A function has a right-hand limit if $f (x)$ approaches $L$ as $x$ approaches $a$ where $x > a .$
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 $x$ approaches $a,$ the branches of the graph will approach the same $y -$ coordinate near $x = a$ 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 $a$ 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.

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: $f (a)$ exists.
- Condition 2: $\lim_{x \to a} f (x)$ exists at $x = a .$
- Condition 3: $\lim_{x \to a} f (x) = f (a) .$
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.

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:
$\frac{f (a + h) - f (a)}{h}$
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.