2.5 The precise definition of a limit  (Page 4/8)

You will find that, in general, the more complex a function, the more likely it is that the algebraic approach is the easiest to apply. The algebraic approach is also more useful in proving statements about limits.

Proving limit laws

We now demonstrate how to use the epsilon-delta definition of a limit to construct a rigorous proof of one of the limit laws. The triangle inequality    is used at a key point of the proof, so we first review this key property of absolute value.

Proof

We prove the following limit law: If $\underset{x\to a}{\text{lim}}f\left(x\right)=L$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=M,$ then $\underset{x\to a}{\text{lim}}\left(f\left(x\right)+g\left(x\right)\right)=L+M.$

Let $\epsilon >0.$

Choose ${\delta }_1>0$ so that if $0<\left|x-a\right|<{\delta }_1,$ then $\left|f\left(x\right)-L\right|<\epsilon \text{/}2.$

Choose ${\delta }_2>0$ so that if $0<\left|x-a\right|<{\delta }_2,$ then $\left|g\left(x\right)-M\right|<\epsilon \text{/}2.$

Choose $\delta =\text{min}\left\{{\delta }_1,{\delta }_2\right\}.$

Assume $0<\left|x-a\right|<\delta .$

Thus,

Hence,

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We now explore what it means for a limit not to exist. The limit $\underset{x\to a}{\text{lim}}f\left(x\right)$ does not exist if there is no real number L for which $\underset{x\to a}{\text{lim}}f\left(x\right)=L.$ Thus, for all real numbers L , $\underset{x\to a}{\text{lim}}f\left(x\right)\ne L.$ To understand what this means, we look at each part of the definition of $\underset{x\to a}{\text{lim}}f\left(x\right)=L$ together with its opposite. A translation of the definition is given in [link] .

Finally, we may state what it means for a limit not to exist. The limit $\underset{x\to a}{\text{lim}}f\left(x\right)$ does not exist if for every real number L , there exists a real number $\epsilon >0$ so that for all $\delta >0,$ there is an x satisfying $0<\left|x-a\right|<\delta ,$ so that $\left|f\left(x\right)-L\right|\ge \epsilon .$ Let’s apply this in [link] to show that a limit does not exist.