### Key Concepts

#### 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 $\underset{x\to \phantom{\rule{0.8em}{0ex}}a}{\mathrm{lim}}f(x)=L,$ 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 $f\left(x\right)$ approaches $L$ as $x$ approaches $a$ where $x<a.$ A function has a right-hand limit if $f\left(x\right)$ 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 $y\text{-}$ 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.

#### 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: $f(a)$ exists.
- Condition 2: $\underset{x\to a}{\mathrm{lim}}f(x)$ exists at $x=a.$
- Condition 3: $\underset{x\to a}{\mathrm{lim}}f(x)=f(a).$

- 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:

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