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I would like to know the precedence of the $\lim$ operator.

For instance, given the following expression:

$$f(x) = \lim_{x \to a} u(x) + v(x)$$

Does the limit apply only to the term?

$$f(x)=\left(\lim_{x \to a} u(x)\right) + v(x)$$

Or does it apply to the entire expression?

$$f(x) = \lim_{x \to a} \left( u(x) + v(x)\right)$$

Potato
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Marcos
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  • Your first expression is very ambiguous. I don't think there's a set precedence. Everyone just uses parenthesis so there's no confusion. – Potato Sep 28 '14 at 01:01
  • So, there are no precedence conventions? You always have to use parenthesis? – Marcos Sep 28 '14 at 01:03
  • I don't think so. Using parenthesis makes everything clear. – Potato Sep 28 '14 at 01:03
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    Typically people do; taking a limit isn't as trivial a thing as doing arithmetic, so generally the context will tip you off. – Milo Brandt Sep 28 '14 at 01:04

1 Answers1

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In most textbooks I've seen the limit operator has higher precedence than addition/subtraction:

$$\lim_{x \to a} u(x) + v(x) \equiv \left(\lim_{x \to a} u(x)\right) + v(x)$$

Where it gets hairy is whether the limit operator has higher precedence than multiplication/division:

$$\lim_{x \to a} u(x) v(x) \stackrel?= \left(\lim_{x \to a} u(x)\right) v(x)$$

I don't think there's an established convention so you would have to guess from context. However, it's usually a bad idea to shadow variables (to reuse the same variable symbol for both the variable in the limit and also another variable outside the limit). So if you wanted to be absolutely clear, it's a good idea to write your equation as:

$$f(x) = \lim_{y \to a} u(y) + v(x)$$

Rufflewind
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