Limit at Infinity and Infinite Limit

7.7.1 Limit at infinity. For $f(x)=1/x$, as x grows without bound (through larger and larger positive values, $x\to\infty$) the value $1/x$ shrinks toward $0$; as x runs through larger and larger NEGATIVE values ($x\to-\infty$) $1/x$ also shrinks toward $0$. So $\lim_{x\to\infty}(1/x)=0$ and $\lim_{x\to-\infty}(1/x)=0$. Formally, f tends to l as $x\to\infty$ if, for every $\epsilon>0$, there is a positive number M such that $|f(x)-l|<\epsilon$ for every x in the domain with $x>M$ (and symmetrically, with $x<-M$, for $x\to-\infty$). The standing technique for a limit at infinity of a ratio of polynomials (or of expressions of the indeterminate form $\infty/\infty$) is to divide numerator and denominator by the highest power of x present, so every remaining term of the form $k/x^p$ ($p>0$) vanishes and only the finite leading-coefficient ratio survives. So for $\lim_{x\to\infty}\dfrac{ax+b}{cx+d}$, dividing by x gives $\dfrac{a+b/x}{c+d/x}\to\dfrac ac$; for a ratio of quadratics, dividing by $x^2$ gives the ratio of the $x^2$-coefficients; and if the numerator's degree is smaller than the denominator's (or vice-versa), the extra powers of x in the leftover factor force the limit to 0 or to $\pm\infty$ respectively. A radical difference such as $\lim_{x\to\infty}\left(\sqrt{x^2+3x}-x\right)$ is first rationalized (multiplying and dividing by $\sqrt{x^2+3x}+x$) to reach $\dfrac{3x}{\sqrt{x^2+3x}+x}$, then divided by x to give $\dfrac{3}{\sqrt{1+3/x}+1}\to\dfrac32$.\n\n7.7.2 Infinite limits. A separate idea: for $f(x)=1/x$ as x approaches the FIXED point $0$ from the right (through small positive values), $1/x$ grows without any bound, written $\lim_{x\to0^+}f(x)=\lim_{x\to0^+}(1/x)\to\infty$; approaching $0$ from the left, $1/x$ plunges without bound, $\lim_{x\to0^-}f(x)\to-\infty$. Because the two one-sided behaviours disagree (one shoots to $+\infty$, the other to $-\infty$), the two-sided lim …