Is it incorrect to substitute the expression of a function by the expression of it's asymptote in a limit to infinity?

real-analysis limits functions

Solution 1:

The fact that $m_1x+b_1$ is an slant asymptote of $f(x)$ means $\displaystyle \lim_{x\to \infty} \frac{f(x)}{m_1x+b_1} = 1$.

So $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=\lim_{x\to\infty}\frac{f(x)}{m_1x+b_1}\cdot \frac{m_1x+b_1}{g(x)} = 1\cdot \lim_{x\to\infty}\frac{m_1x+b_1}{g(x)}$$

Do the same for $g(x)$

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