When can limits to both infinities exists and differ?

Question

What property of a function required to make: $$\lim_{x\to\infty} f(x)=M \neq \lim_{x\to-\infty} f(x)=N$$

Such as: $$f(x)=\frac{\sqrt{2x^2+1}}{3x-5}$$ $$\lim_{x\to\infty}f(x) = \frac{\sqrt{2}}{3}$$ But$$\lim_{x\to-\infty}f(x) = -\frac{\sqrt{2}}{3}$$

Answer 1

There is nothing special in this condition, for example for all functions in the form with $a,c \neq 0$

$$f(x)=\frac{a|x|+b}{cx+d}$$

we have that

$$\lim_{x\to\infty}f(x) = \frac{a}{c}\quad \neq \quad \lim_{x\to-\infty}f(x) = -\frac{a}{c}$$

and we can construct many of these examples.