Eroxl's Notes
Limit

The limit of a function describes the value that function approaches as its input approaches a certain value.

In formulas, the limit of a function is usually written as

This formula describes the limit of as approaches the constant

Note

It does not matter what the output of a function is or even if it exists for a limit to be taken.

Limits can also be taken from only one side meaning as the input variable approaches from either the left or right side of the graph it approaches a certain value. These limits are called one-sided limit and are useful when the limit at a point does not exist.

Limitations

For a limit to exist both sides of the value the input approaches must agree for example in the following function there does not exist a limit at

Y axisX axis00-1-1112233445555Expression 1Expression 2Expression 3Expression 4

Using one-sided limits this limitation can be described as the limit of a function at a point does not exist if the left and right sided limits are not equal at that point.

Examples

Y axisX axisExpression 1Expression 2

Notice how the limit at is even though the function is not defined at . This is > because the function approaches as approaches .

Notice how the limit at is even though the function is defined to be at . This is because the function approaches as approaches .

Y axisX axisExpression 1Expression 2Expression 3

Properties

Sum Rule

Product Rule

Quotient Rule

Constant Multiple Rule

Where is any constant number

Power Rule

Calculating

Graphically

One way to calculate a limit is to graph the function . The limit of is the value that the graph approaches as the input value. This can be seen graphically by looking at the values of the function as the input gets closer to the desired value.

For example given the following function the limit as approaches can be calculated using the following graph

Y axisX axis00-2-2-1-111223344-2-22244Expression 1Expression 2

Using this graph we can estimate that the function was probably heading towards at . This means we can write the limit as follows

Numerically

Another way to calculate a limit is to create a table of values that show the function evaluated at numbers that are close to the desired limit. The limit is the value that the function approaches as these numbers get closer to the desired limit.

For example given the following function the limit as approaches can be calculated using the following table
0.5405
2.90 0.5263
2.95 0.5128
3.00 undefined
3.05 0.4878
3.10 0.4762
3.15 0.4651

Using this table we can estimate that the function was probably heading towards at . This means we can write the limit as follows

Symbolically

We can also calculate the limit symbolically using algebraic techniques. This involves trying to simplify the function into an expression where the desired limit is easily visible.

Methods

Direct Substitution

For functions that are continuous at then the limit . Most limits solved symbolically re-arrange the equation such that direct substitution can be applied.

Example

Given the function , the limit as approaches .

Factoring and Simplifying

The most common technique for calculating limits is by factorizing the function and removing any discontinuities.

Example

Given the function , the limit as approaches .

Rationalizing

Sometimes rationalizing either the numerator or the denominator can help. Usually this only works for questions with radicals.

Example

Given the function , the limit as approaches

Alternative Approaches

L'Hôpital's Rule

L'Hôpital's rule is a mathematical theorem that makes it possible to evaluate limits of an indeterminate form.

The Squeeze Theorem

The squeeze theorem is a mathematical theorem for finding the limit of a function trapped between two other functions.