Adding and Subtracting Limits Examples

Given that ,

what is ?

The first thing we do is break the limit into several pieces:

Now we pull out constants, and evaluate the limits of x and 4:

Since ,

.

What is ,

given that ?

Time to break out the addition rule for limits.

The limit of the sum is the sum of the limits:

.

We know

,

therefore

.

Solving the equation to get the limit all by itself on one side, we find

.

Find ,

given that .

Since the limit of the difference is the difference of the limits,

We know

,

so putting it all together we find

.

Rearranging,

.

Assuming that ,

what is ?

Since all the necessary limits exist for the limit of a sum (or difference) to be the sum (or difference) of the limits,

.

Pulling out the constants, and then using our rules about limits of x and of constants,

.

Putting things together, we now have the equation

.

Rearranging gives us

,

which means

.