# 1.6: Properties of Limit

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- 5677

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:

Therefore:

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.

Now, evaluate

Based on the properties of limits, the limit can be evaluated in the following steps:

Therefore:

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.

## Examples

### Example 1

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:

### Example 2

Evaluate

where f(x) is the rational function

### Example 3

Find the following limit if it exists:

Let’s apply the basic quotient rule to evaluate this limit.

Therefore:

Again, the limit result is the same as using direct substitution of x=−4 in the function.

## Review

## Vocabulary

Term | Definition |
---|---|

indeterminate |
In mathematics, an expression is indeterminate if it is not precisely defined. There are seven indeterminate forms: ^{0}/_{0},0⋅∞,∞∞,∞−∞,0^{0},∞^{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. |

## Additional Resources

Practice: Properties of Limits