# Calculus 2: Sequence Limits Question to the power n?

1. May 3, 2013

### raaznar

Calculus 2: Sequence Limits Question to the power n??

1. The problem statement, all variables and given/known data
Find the limits (if it exists) to decide which sequences, whose nth term is given below.

2. Relevant equations
$(\frac{3^{n}-4^{n}}{3n^{2}+4^{n}+7})$

3. The attempt at a solution
I've done a few of these but as Soon as the constants were raised to the n. It pretty much stumped me. I haven't done stuff like this yet and I'm sort of self teaching.

I would appreciate some direction rather than an answer. Thanks!

Last edited: May 3, 2013
2. May 3, 2013

### Zondrina

Notice the quantity in your numerator is always smaller than zero.

Notice your denominator is growing without bound.

So what does the whole thing tend to as n → ∞?

3. May 3, 2013

### HallsofIvy

Staff Emeritus
As n grows, exponentials, numbers to the n power, will increase faster than n to a power. And, of course, a larger base will increase faster than a smaller base. That means that $$4^n$$ will 'dominate' here.

Divide each term in numerator and denominator by $4^n$ to get
$$\frac{\left(\frac{3}{4}\right)^n- 1}{3\frac{n^2}{4^n}+ 1+ \frac{7}{4^n}}$$

Now, what does each of those fractions go to?

4. May 3, 2013

### raaznar

0? I got 0 to start of with but the answer is -1. If that's what you are implying. I have no idea how to mathematically arrive at the answer -1.

5. May 3, 2013

### raaznar

Negative 1. Got it! I've never done the 'Dominate Term' Approach. Should this be the first approach to consider when doing limits of sequences?

What's the first things I should consider when I approach questions like this in the future? (Whats the checklist to look out for)

Thanks man by the way!