The textbook I am using has come up with this method to prove why in infinite limits, the term with the highest degree will dominate:
It then goes on to say that at infinity, 1000/3x and all the other terms will be virtually 0. Therefore, the function will approach 1 (since it it lim x->infinity (1-0+0-0).
However, my problem with this, is for instance, take a function x+2/x-1. On the logic of the above, when we break it up, it is (x+2)*(1/x-1). At infinity, 1/x-1=0. So therefore, the limit is 0. But the limit of x+2/x-2 is actually 1.
We see this again we we take a function sin(x)*1/x. Again, at infinity, the limit should be 0, but it's not.
So what can explain this contradiction? Why is the above proof still right in these situations despite the examples below? How can we know that the proof above is universally true? And why is my logic wrong? Why is is x+2/x-1 not equals to 0?
Best Answer
Applying the same method to $\frac{x+2}{x-1}$, what we get is\begin{align}\lim_{x\to\infty}\frac{x+2}{x-1}&=\lim_{x\to\infty}\frac{(x-1)+3}{x-1}\\&=\lim_{x\to\infty}1+\frac3{x-1}\\&=1.\end{align}
And the limit $\lim_{x\to\infty}\frac{\sin x}x$ is $0$.