Extension: Uniqueness of Limits at Infinity

First we should ask ourselves why we need to show that limits are well defined.

Yeah it is pretty "obvious," since we have an intuitive notion of what a limit is.

What does it mean when we say, "limit of a function is well defined?"

Consider the function $f(x) = \sin(x)\cdot (e^{-x} + 0.2).$

x

y

Using some intuition for what a limit is, this function gets close to both $+.2$ and $-.2$ at infinity so we can say $\lim_{x\to\infty} f(x)$ is both $.2$ and $-.2,$ right?

Why can't we say this function has a limit at infinity?

As we saw above, $f(x)$ does not have a limit at infinity so we cannot actually assign a number to $\lim_{x\to\infty} f(x).$

This is where our well-definedness of limits is important. If we have that $\lim_{x\to a}f(x)$ exists, then the definition implies that there will be one (and only one) number equal to $\lim_{x\to a}f(x).$

Good, now let's see a proof!

Extension: Uniqueness of Limits at Infinity

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