Limit to infinity rule for fractions?

Question

I am reading a book and it says to solve limits to infinity with a fraction such as:

$$\frac{5X^2 + 8X - 3}{3X^2 + 2}$$

We divide the numerator and denominator by the highest power of X in the DENOMINATOR so in this case it is $X^2$. I get this helps simplify the equation, but what is to prevent someone from dividing by a higher power like $X^3$? All components would evaluate to 0.

Is there another rule for limits that I am not aware of?

Thanks!

Answer 1

Yes we can divide by $X^3$ but we obtain

$$\frac{5X^2 + 8X - 3}{3X^2 + 2}=\frac{\frac 5 X + \frac 8{X^2} - \frac 3{X^3}}{\frac 3X + \frac2{X^3}}$$

which is again an indeterminate form.

In general, to avoid that this happens, the standard way is to factor out the dominating term from the numearator and the denominator.

Answer 2

The most dominant term in num is $5x^2$ and the most dominant term in den is $3x^2$ when $x$ is very large. So the required limit is $$L=\lim_{x\to \infty} \frac{5x^2}{3x^2}=\frac{5}{3}.$$