# #lim_(x->oo)(3^x)/(x2^x)=#?

##### 1 Answer

The fractional limit increases without bound as

#### Explanation:

Let's shuffle things around so that there's only one exponential:

The above representation of the limit has both the numerator and denominator growing without bound as *faster* than the other? Or perhaps, does the ratio of their growing speeds approach a finite value?

We can answer this question by finding the *rates* that both sides of the fraction approach infinity, and equating the original limit to **the limit of the ratio of these rates.** In other words, we are going to find their derivatives.

This is called **L'Hôpital's Rule**, and you can use it whenever you have a fractional limit that approaches

Continuing, we have

The multiplier

Now, what's left inside our limit? It's an exponential function of

So we get

Therefore,

## Bonus:

In general, exponential functions (like