<s>[INST] The rule for evaluating limits of rational functions by dividing the coefficients of highest powers I have a Limit problem as below:
Connor claims: " $\lim_{x\to \infty} \left(\frac{6x^2 + 7x +3}{2x^3 + x^2 -2x -1}\right) = 3$ because my high school calculus teacher told us the limit of ratio of polynomials is always the quotient of the coefficients of the highest power terms"
If correct, show in detail how to use algebra and the limit theorems to evaluate this limit and get the same answer
If wrong,
1, use algebra and limit theorems to correctly evaluate the limit and
2, write a paragraph that explain why Connor shouldn't expect the rule he remember from high school to work in this particular problem
I am able to correctly evaluate the limit ( lim = 0) but since my first language is not English, I don't understand what Connor claimed and how to explain it. Can anyone help? Thanks in advance
 [/INST] $$\lim_{x\to \infty} \left(\frac{6x^2 + 7x +3}{2x^3 + x^2 -2x -1}\right)=$$

The leading term in the denominator of $\frac{6x^2 + 7x +3}{2x^3 + x^2 -2x -1}$ is $x^3$. Divide the numerator and denominator by this:

$$\lim_{x\to \infty} \left(\frac{\frac{6}{x}+\frac{7}{x^2}+\frac{3}{x^3}}{2+\frac{1}{x}-\frac{2}{x^2}-\frac{1}{x^3}}\right)=$$

Te expressions $\frac{3}{x^3},\frac{7}{x^2},\frac{6}{x},-\frac{1}{x^3},-\frac{2}{x^3},\frac{1}{x}$ all tend to zero as $x$ approaches $\infty$:

$$\frac{0+0+0}{2+0-0-0}=\frac{0}{2}=0$$
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