# Limits of Exponential Functions

Everyone has their limit; logs and exponents are no different.

Let's look at the exponential function *f *(*x*) = 4^{x}. No matter what value of *x* you throw into it, you can never get *f *(*x*) to be negative or zero. (How optimistic of it.) Try a few:

4^{2} = 16

4^{3} = 64

4^{4} = 256

4^{0} = 1

4^{-2} = ^{1}/_{16}

That's reflected in the function's graph, too. It never dips below *y* = 0.

### Sample Problem

What's an exponential function that contains the following set of points? Hint: remember that anything to the power of 0 becomes 1.

{(2, 18), (0, 2), (3, 54)}

Because any base raised to the power of 0 equals 1, we can easily figure out the coefficient. The point (0, 2) means that *y* is 2 when *x* is 0, or *f* (0) = 2. Throw that bad boy into the mix and solve:

*f* (*x*) = *Cb ^{x}*

2 =

*Cb*

^{0}

2 =

*C*(1)

*C*= 2

Now that you know *C*, use it to find the base. We'll use the point (2, 18), which means *f* (2) = 18:

*f* (*x*) = 2*b ^{x}*

18 = 2

*b*

^{2}

9 =

*b*

^{2}

*b*= 3

Remember, we can't have a negative base. That's why we're only using the positive square root of 9, even though it's technically ±3.

So the exponential function we need is *f* (*x*) = 2(3)* ^{x}*. Let's double-check that it works with our third point, (3, 54).

*f* (*x*) = 2(3)^{x}*f *(3) = 2(3)^{3} = 2(27) = 54

Yep, we're in the clear.