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# Class Contents

## Limits of functions

Example:

Example:

Example:

Note that we can use Proposition 4.2, together with our results on the arithmetic of convergent sequences
( Theorem 3.9) to prove Theorem 4.1 quickly.
If `(x_n)` converges to `a` in `D` but is never equal to `a`, then `f(x_n)+g(x_n)`, `f(x_n)g(x_n)`, and `f(x_n)/g(x_n)`
converge to `A+B`, `AB`, and `A/B` respectively, by
Theorem 3.9, and one half of Proposition 4.2.

But then the other direction in Proposition 4.2 shows that, since `(x_n)` was an arbitrary sequence, `f(x)+g(x)`, `f(x)g(x)`, and `f(x)/g(x)` converge to the same limits.

**Proof.** Use Proposition 4.2 and Lemma 3.3.

Example:

We can reprove an earlier example much more easily now:

Simply take a sequence `p_n` of rationals converging to
`a` (without ever reaching it) and a sequence `x_n` of
irrationals converging to `a` without ever reaching it. Then
`f(p_n)=1` and `f(x_n)=0` for all `n`, and so the
sequences cannot converge to a common limit. It follows from Proposition
4.2 that the limit of the function does not exist.