12.2 Finding limits: properties of limits

Consider the rational function    

The function can be factored as follows:

Does this mean the function $f$ is the same as the function $g(x)=x+1?$

The answer is no. Function $f$ does not have $x=7$ in its domain, but $g$ does. Graphically, we observe there is a hole in the graph of $f\left(x\right)$ at $x=7,$ as shown in [link] and no such hole in the graph of $g\left(x\right),$ as shown in [link] .

So, do these two different functions also have different limits as $x$ approaches 7?

Not necessarily. Remember, in determining a limit    of a function as $x$ approaches $a,$ what matters is whether the output approaches a real number as we get close to $x=a.$ The existence of a limit does not depend on what happens when $x$ equals $a.$

Look again at [link] and [link] . Notice that in both graphs, as $x$ approaches 7, the output values approach 8. This means

Remember that when determining a limit, the concern is what occurs near $x=a,$ not at $x=a.$ In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.

Finding the limit of a sum, a difference, and a product

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits    , which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.