You're familiar with the idea of a limit of a function, and that some limits are computed using numerical and graphical methods. Limits can also be evaluated using the properties of limits. How would you find
without using a graph or using a table of values?
Properties of Limits
Let’s begin with some observations about limits of some simple functions. Consider the following limit problems:
We note that each of these functions is defined for all real numbers. If we apply our intuition for finding the limits we conclude correctly that:
The above results can be encapsulated in the following limit properties:
Basic Limit Properties:
Many functions can be expressed as the sums, differences, products, quotients, powers and roots of other more simple functions. The following properties are also useful in evaluating limits:
More Basic Limit Properties:
Knowing these properties, allows evaluation of the limits of a wide range of functions.
Take the problem:
Based on the properties above, the limit can be evaluated in the following steps:
Note that the application of the basic limit properties results, in this case, in a limit value that is the same as direct substitution of x=2 in the function.
Based on the properties of limits, the limit can be evaluated in the following steps:
Again, note that the application of the basic limit properties results, in this case, in a limit value that is the same as direct substitution of x=2 in the function.
Earlier, you were asked to find
without using a graph or a table of values. This is a case where direct substitution to evaluate the limit gives the indeterminate form 0/0. But, reducing the fraction and then applying the basic limit properties above, we can evaluate the limit:
where f(x) is the rational function
Find the following limit if it exists:
Let’s apply the basic quotient rule to evaluate this limit.
Again, the limit result is the same as using direct substitution of x=−4 in the function.
To see the Review answers, open this PDF file and look for section 2.3.
|indeterminate||In mathematics, an expression is indeterminate if it is not precisely defined. There are seven indeterminate forms: 0/0,0⋅∞,∞∞,∞−∞,00,∞0, and 1∞.|
|limit||A limit is the value that the output of a function approaches as the input of the function approaches a given value.|
Practice: Properties of Limits