# At a Glance - Powers and Roots of Limits

Limits are pretty powerful. They're kind of the big idea of calculus. Throughout calculus we'll see that no matter what we're doing, there's a limit or two lurking somewhere.

The purpose of this reading isn't to totally fangirl over limits, though. Instead we'll be talking about what happens when we take a limit involving a function raised to some power. As it turns out, there's a property to help us with this very situation.

## Power Property

If   exists, and p is any real number, then

The limit of a function that's being raised to some power is the limit of that function raised to the same power. All there is to it.

### Sample Problem

If   then

Got it? Just pull the power out of the limit.

#### Example 1

 What is  if ?

#### Example 2

 What is  if ?

#### Example 3

 What's  if ?

#### Exercise 1

Evaluate the limit.

#### Exercise 2

Evaluate the limit.

#### Exercise 3

Evaluate the limit.

#### Exercise 4

Evaluate the limit.

• , assuming that

#### Exercise 5

Evaluate the limit.

• , assuming .

#### Exercise 6

Find all possible values for the specified limit.

• , assuming .

#### Exercise 7

Find all possible values for the specified limit.

• , assuming .

#### Exercise 8

Find all possible values for the specified limit.

• , assuming

#### Exercise 9

Find all possible values for the specified limit.

• , assuming .

#### Exercise 10

Find all possible values for the specified limit.

• , assuming