Definition of Limits at Infinity

Let $L$ be a real number. The statement

$$\lim_{x\to\infty} f(x)=L$$

means that for each $\epsilon >0$ there exists an $M>0$ such that

$$\left\lvert f(x)-L \right\rvert <\epsilon \quad x>M.$$

The statement

$$\lim_{x\to -\infty} f(x)=L$$

means that for each $\epsilon >0$ there exists an $N<0$ such that

$$\left\lvert f(x)-L \right\rvert <\epsilon \quad x<N.$$