1. ## Slant Asymptote

Given the function $\displaystyle f(x)=\frac{3x^4+4}{x^3-3x}$ I have found, by long division, that there is a slant asymptote, y=3x.
Further, I wanted to find the same result by the classic limit definition.

$\displaystyle \lim_{x \mapsto \infty}\left [ \frac{3x^4+4}{x^3-3x} \right ]$

$\displaystyle \lim_{x \mapsto \infty}\left [ \frac{x^4(1+\frac{4}{x^4})}{x^4(\frac{1}{x}-\frac{3}{x^3})} \right ]$

$\displaystyle \lim_{x \mapsto \infty}\left [ \frac{1+\frac{4}{x^4}}{\frac{1}{x}-\frac{3}{x^3}} \right ]=\frac{1+0}{0-0}$

I ended up in an indeterminate form.The result should had be the same!What's wrong?

2. ## Re: Slant Asymptote

$\displaystyle \frac{1}{0}$ is not an indeterminate form.

3. ## Re: Slant Asymptote

I made a mistake in the simplification.The final result is $\displaystyle \frac{3}{0}$. The problem is division by zero!

4. ## Re: Slant Asymptote

Originally Posted by joaofonseca
I made a mistake in the simplification.The final result is $\displaystyle \frac{3}{0}$. The problem is division by zero!
which means the limit does not exist.

5. ## Re: Slant Asymptote

The limit exist!Look at the graph:

6. ## Re: Slant Asymptote

I'm afraid you are confused ... the limit as $\displaystyle x \to \infty$ will determine a horizontal asymptote, but not a slant asymptote.

After you do the long division, you get an expression of the form $\displaystyle y = mx + b + r(x)$, where $\displaystyle r(x)$ is a remainder that tends to 0 as $\displaystyle x \to \infty$ , telling you that the function value approaches the line $\displaystyle y = mx+b$