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I'm a bit confused about the idea of adding limits when they do not exist. I'm reading a book on Calculus, and it states that $ \lim_{x \to a} \ [f(x) + g(x)] $ exists in the following cases:

  1. Both $ \lim_{x \to a} f(x) $ and $ \lim_{x \to a} \ g(x) $ exist.
  2. Neither $ \lim_{x \to a} f(x) $ nor $ \lim_{x \to a} \ g(x) $ exists.

If that's the case, how can the following limit exist:

$$ \lim_{x \to 0} \csc(x) \ + \lim_{x \to 0} \cot(x) \ = \lim_{x \to 0} \frac{1+\cos(x)}{\sin(x)} $$

Thanks!

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    $\begingroup$ Who says they exist? $\endgroup$ – J. W. Tanner Mar 19 at 0:10
  • $\begingroup$ No one said they exist, I just randomly came up with an example that, I think, contradicts that second argument - the limit of the sum of two functions exists if neither the limit of the first function nor the second exists. Am I misreading this somehow? $\endgroup$ – nullbyte Mar 19 at 0:20
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    $\begingroup$ null, if you read your book a bit more carefully, you will find some context, some restrictions on $f,g,a$ that you have not yet told us $\endgroup$ – Will Jagy Mar 19 at 0:21
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    $\begingroup$ lim $f+g$ may exist if lim $f$ and lim $g$ don't, but there are plenty of examples where lim $f$ and lim $g$ don't exist and lim $f+g$ doesn't either $\endgroup$ – J. W. Tanner Mar 19 at 0:30
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    $\begingroup$ It's possible what the book said (or meant to say) is that if the LHS limit (i.e., the limit of the sum) exists, then either both limits on the RHS exist or neither of them exists. $\endgroup$ – Barry Cipra Mar 19 at 0:56
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You must have misunderstood what the book says.

According to the algebraic limit theorem, $\lim_{x\to a}[f(x)+g(x)]=\lim_{x\to a}f(x)+ \lim_{x\to a}g(x),$

provided the limits on the right side exist.

It is possible for $\lim_{x\to a} [f(x)+g(x) ]$ to exist when $\lim_{x\to a}f(x)$ and $ \lim_{x\to a}g(x)$ do not,

but there are plenty of examples (such as the one you gave in the question)

where $\lim_{x\to a}f(x)$ and $ \lim_{x\to a}g(x)$ do not exist and $\lim_{x\to a} f(x)+g(x)$ does not either.

Perhaps the book was trying to say that you cannot have the situation where

$\lim_{x\to a} [f(x)+g(x) ]$ and $\lim_{x\to a}f(x)$ exist but $ \lim_{x\to a}g(x)$ does not.

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