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Math and science::Analysis::Tao::09. Continuous functions on R

Limit laws for functions, proposition

Combining the rudimentary definitions of arithmetic operations on functions with the weildy definition of function limits gives us the limit laws of functions.

As words, they are as follows:

Let \( X \) be a subset of \( \mathbb{R} \), let \( E \) be a subset of \( X \), let \( x_0 \) be an adherent point of \( E \) and let \( f : X \to \mathbb{R} \) and \( g : X \to \mathbb{R} \) be functions. Suppose that \( f \) has a limit \( L \) at \( x_0 \) in \( E \), and \( g \) has a limit \( M \) at \( x_0 \) in \( E \), then:

  • \( f+g \) has a limit \( L + M \) at \( x_0 \) in \( E \)
  • \( f-g \) has a limit \( L - M \) at \( x_0 \) in \( E \)
  • \( \max(f, g) \) has a limit \( \max(L, M) \) at \( x_0 \) in \( E \)
  • \( \min(f, g) \) has a limit \( \min(L, M) \) at \( x_0 \) in \( E \)
  • \( fg \) has a limit \( LM \) at \( x_0 \) in \( E \)
  • \( cg \) has a limit \( cL \) at \( x_0 \) in \( E \) (assuming if \( c \) is a real number)
  • \( f/g \) has a limit \( L / M \) at \( x_0 \) in \( E \) (assuming \( g(x) \ne 0 \text{ for all } x \in E \) and \( M \ne 0 \))


Each of these can be expressed more informally (losing reference to sets \( X \), \( \mathbb{R} \) and \( E \) and dropping the specification of restrictions) like so (just one example here):

\[ lim_{x \rightarrow x_0}(f/g)(x) = \frac{\lim_{x \rightarrow x_0} f(x)}{\lim_{x \rightarrow x_0} g(x)} \]

Where does the effort lie in these propositions

I find it easier to think about the above equation going from right to left, in the same direction as the worded version above proceed. One way of thinking about these propsotions is that when both \( f \) and \( g \) satisfy the criteria to have a limit at \( x_0 \) in \( E \), then so too does the function \( (f+g) \). The value of the limit of \( (f+g) \) at \( x_0 \) in \( E \) must be the sum of the limits of \( f \) and \( g \). The latter idea follows intuatively once the former idea (that \( (f+g) \) must have a limit) is accepted.

Source

p225