Is the Polynomial Limit Theorem Accurate and Comprehensive?

[Orion1]

My Calculus professor has indicated a 'shortcut' in determining polynomial fraction limits, I am inquiring if this identity is correct, and how comprehensive is this 'theory'?

Polynomial Limit Theorem:
$$\lim_{x \rightarrow \infty} \frac{ax^2 - x + 2}{bx^2 - 1} = \frac{a}{b}$$

 

[Tx]

My Calculus professor has indicated a 'shortcut' in determining polynomial fraction limits, I am inquiring if this identity is correct, and how comprehensive is this 'theory'?
Polynomial Limit Theorem:
$$\lim_{x \rightarrow \infty} \frac{ax^2 - x + 2}{bx^2 - 1} = \frac{a}{b}$$

Just divide through by the highest power, then the limit becomes A/B as x -> oo.

 

[VietDao29]

You can also expand your Polynomial Limit Theorem like this:
Let $m , \ n \in \mathbb{Z ^ +}$
If m < n:
$$\lim_{x \rightarrow \infty} \frac{\sum \limits_{i = 0} ^ m a_i x ^ i}{\sum \limits_{k = 0} ^ n b_k x ^ k} = 0 \quad (a_m, b_n \neq 0)$$
If m > n:
$$\lim_{x \rightarrow \infty} \frac{\sum \limits_{i = 0} ^ m a_i x ^ i}{\sum \limits_{k = 0} ^ n b_k x ^ k} \quad (a_m, b_n \neq 0)$$ it does not have a limit.
If m = n:
$$\lim_{x \rightarrow \infty} \frac{\sum \limits_{i = 0} ^ m a_i x ^ i}{\sum \limits_{k = 0} ^ n b_k x ^ k} = \frac{a_m}{b_n} \quad (a_m, b_n \neq 0)$$