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I recently was reading through my James Stewart calculus book and learnt the limit laws . Take for example the sum rule if the limit of 2 functions exist then the sum of their limits is equal to the limit if their sums.this we can use to prove the sum rule for derivatives. However when they say exists what exactly do they mean for example does the limit need to be a real number.for example if the limit of some function is $\lim_{x \to a} f(x)$=$+\infty$ Does the limit not exist and hence we cannot use the sum rule ?

Bernard
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The homeschooler
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  • Depends. Some authors say 'exists and is finite' and other just say 'exists' even when the limits/derivatives are $\pm \infty$. – Kavi Rama Murthy Dec 25 '20 at 10:15
  • Would that imply that I could use the sum rule ? – The homeschooler Dec 25 '20 at 10:17
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    If $f(x) \to \infty$ and $g(x) \to \infty$ then $f(x)+g(x) \to \infty$. But if $f(x) \to \infty$ and $g(x) \to -\infty$ then we cannot say anything about the limit of $f(x)+g(x)$. – Kavi Rama Murthy Dec 25 '20 at 10:18
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    Best to regard $\infty$ as a symbol of unbounded growth, rather than an actual number. – user2661923 Dec 25 '20 at 10:21
  • I clearly would prefer that $\infty$ (or $-\infty$) is not considered to be an existing limit. Strange that there is no consens about the terminology here. – Peter Dec 25 '20 at 10:57

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As some people in the comments have said, there really lacks a consensus about whether a limit exists if it is equal to $\infty$. Some will say "it diverges" if it is unbounded, others will say "it converges to $\infty$", saving "diverges" for when it oscillates or something. As far as the sum rule goes, if you are adding $\infty$ to any constant then it works, but if you are adding $\infty$ to $-\infty$, then you have indeterminate and should try calculating the derivative another way.

Neptune
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