# Limit evaluation at infinity and 0 by intuition

$$\lim_{x\to\infty,y\to\infty} \frac{x}{4y}$$ $$\lim_{x\to0,y\to0} \frac{x}{4y}$$

At $0$, since $x$ and $y$ both are limiting to $0$, they are equal, and the limit comes out to be $1/4$.

At infinite, $x$ is tending to infinite and so is $y$. $4$ times a very very large quantity should be $4$ four times more than another large quantity (approximately). Since, we're considering the limits, they should be equal. Therefore, limit should be $1/4$.

Where am I wrong?

Because when $x$ and $y$ both approaching $0$ and $\infty$, they are not necessarily approching $0$ (or $\infty$) at the same speed.
For instance, if you take $x_n = \frac{1}{n^2}$ and $y_n = \frac{1}{n}$, it is obvious that $\frac{x_n}{4y_n}$ will tend to $0$ as $n \rightarrow \infty$.
But if you take $x_n = y_n = \frac{1}{n^2}$, $\frac{x_n}{4y_n}$ will tend to $1/4$ as $n \rightarrow \infty$.
These two different limits ($0$ and $1/4$) prove that $\lim_{x\to0,y\to0} \frac{x}{4y}$ does not exist.