How to compute the limit of a rational function at infinity?

Question

I am unable to compute the limit, anyone can help ? I only understand some basic knowledge of limit .

Answer 1

Rewrite this as $$\lim_{x \to \infty} \frac{\frac{6x}{x^2} - \frac{7}{x^2}}{\frac{10x^2}{x^2} - \frac{10x}{x^2} + \frac{3}{x^2}}$$

Notice how we divided out the largest power in the denominator. This is the usual strategy for dealing with infinite limits like this. Then as you can see, every term goes to $0$ as $x \to \infty$, because if you have a constant over a variable that goes to infinity, then the fraction is getting smaller and smaller, so the limit is $0$ (the exception here is the $\frac{10x^2}{x^2}$, but that doesn't matter, because the top part of the overall fraction goes to $0$). After you evaluate this, you get $$\frac{0}{10} = 0$$

If you're wondering why we divide out by the largest power in the denominator: it's because what matters when we evaluate infinite limits is the largest power. As $x$ grows larger and larger, the smaller terms play a smaller role in things. If we divide every term by the largest power like that, then it allows us to cancel out the terms that don't make a huge difference in the long run, because they go to $0$.

Answer 2

To expand on Alex's answer. For any simple limit like this where you have

$$\lim_{x \to \infty} \frac{h(x)}{g(x)}$$

Where $h(x)$ and $g(x)$ are simple polynomials, the limit will be equal to the coefficients of the highest order terms.

So in your case, the highest order term is $x^2$. So this limit is equal to $\frac{0}{10}=0$ as Alex said.

If you instead had: $$\lim_{x \to \infty} \frac{5x^2+6x-7}{10x^2-10x+3}$$

The answer would be $\frac{5}{10}=\frac{1}{2}$