Limit of a rational function for different values of K (check)

[MurdocJensen]

Find the limit of (4^kx + 9)/(4^5x + 3) as x->inf, when K=5, K<5, K>5.

Answer: I divided each term by the highest power of 4, and was able to come up with 0 (K<5), 1 (K=5), inf (K>5). I'm pretty sure these are correct.

Was this the right method? Am I good to assume that dividing by the highest power of x is similar in some ways to dividing by the highest power of some common base, which is what I did here?

Let me know if I can explain myself better.

 

[gb7nash]

Find the limit of (4^kx + 9)/(4^5x + 3) as x->inf, when K=5, K<5, K>5.

Answer: I divided each term by the highest power of 4, and was able to come up with 0 (K<5), 1 (K=5), inf (K>5). I'm pretty sure these are correct.

Was this the right method? Am I good to assume that dividing by the highest power of x is similar in some ways to dividing by the highest power of some common base, which is what I did here?

Let me know if I can explain myself better.

You are indeed correct. If you want to be more rigorous for K = 5, K > 5, and 0 <= k < 5 use the following trick:

$$\frac{4^{kx} + 9}{4^{5x} + 3} = \frac{4^{kx} + 9}{4^{5x} + 3} \ * \frac{\frac{1}{4^{kx}}}{\frac{1}{4^{kx}}} = \frac{1 + 9*4^{-kx}}{4^{(5-k)x} + 3*4^{-kx}}$$

For k < 0, you can use what you originally typed (do you know why?)

 

[MurdocJensen]

Thanks.

And for K < 0, I can use what I originally typed because the terms with x in their exponent will tend to zero as x -> inf.

And how do you type those operations out? They look so pretty.

 

[micromass]

Thanks.

And for K < 0, I can use what I originally typed because the terms with x in their exponent will tend to zero as x -> inf.

Yes, for K<0, we have that $4^{Kx}\rightarrow 0$. And thus

$$\frac{4^{Kx}+9}{4^{5x}+3}\rightarrow 0$$

as the denominator will tend to infinity.

And how do you type those operations out? They look so pretty.

Click on the equations to see the code!

 

[gb7nash]

Latex. There's a pretty decent tutorial on it here:

http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/

 

[MurdocJensen]

micromass: I was pretty general (and wrong) with my reply. Thanks for the clarification.