13.2 Standard Limits

(i).: \(\lim \limits _{x\to 0}x=0\).
(ii).: \(\lim \limits _{x\to \infty }x=\infty \); the limit does not exist as a real number, and writing \(\infty \) records how it fails — by growing without bound. \(\infty \) is not a real number and cannot be substituted.
(iii).: \(\lim \limits _{x\to \infty }\frac {1}{x}=0\).
(iv).: \(\lim \limits _{x\to 0}\frac {1}{x}\) does not exist, since \[\lim _{x\to 0^{-}}\frac {1}{x}=-\infty \qquad \text {and}\qquad \lim _{x\to 0^{+}}\frac {1}{x}=+\infty .\]

Note 13.10. Item (iv) needs care. The two one-sided limits disagree — they run off in opposite directions — so the two-sided limit does not exist, and writing \(\lim \limits _{x\to 0}\frac {1}{x}=\infty \) is wrong. Contrast \(\frac {1}{x^{2}}\), where both sides tend to \(+\infty \) and one may write \(\lim \limits _{x\to 0}\frac {1}{x^{2}}=+\infty \) — still a limit that fails to exist, but failing in one describable way.

If \(\lim \limits _{x\longrightarrow a}f(x)=L\) and \(\lim \limits _{x\longrightarrow a}g(x)=K.\)

Then

1.: \(\lim \limits _{x\longrightarrow a}(f(x)+g(x))=\lim \limits _{x\longrightarrow a}f(x)+\lim \limits _{x\longrightarrow a}g(x)=L+K\)
2.: \(\lim \limits _{x\longrightarrow a} (f(x)\cdot g(x))=L\cdot K\)
3.: If \(K\neq 0\). Then \(\lim \limits _{x\longrightarrow a}\left (\frac {f(x)}{g(x)}\right )=\frac {\lim \limits _{x\longrightarrow a}f(x)}{\lim \limits _{x\longrightarrow a}g(x)}=\frac {L}{K}\).
4.: If \(K\) is a constant the \(\lim \limits _{x\longrightarrow a} K=K\).

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