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Prove that $\lim_{x\to a}\frac{1}{x}=\frac{1}{a}$ for any $a$ in $(-\infty,0)$ .

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Lesson Parent:

Function Limits III: The Gory Details

Problem:

Prove that
$\lim_{x\to a}\frac{1}{x}=\frac{1}{a}$
for any $a$ in $(-\infty,0)$ .

Answer:

It is true that $\lim_{x\to a}\frac{1}{x}=\frac{1}{a}$ for any $a$ in $(-\infty,0)$ .

Solution:

So any $a$ in <Sign in to see all the formulas> means <Sign in to see all the formulas>.

So from our definition of function limits we need to prove that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas>, then <Sign in to see all the formulas>.

Now recall that <Sign in to see all the formulas> means that <Sign in to see all the formulas> or <Sign in to see all the formulas>. But $x$ cannot be zero because $1/x$ when $x=0$ is undefined. So let's make sure that $x$ is always negative in our proof by making sure that <Sign in to see all the formulas>. In other words, let's choose a minimum value for <Sign in to see all the formulas>. Anything will do so long as <Sign in to see all the formulas>. Remember $a$ is negative so <Sign in to see all the formulas> is positive and we can write <Sign in to see all the formulas>. Since $a$ is any arbitrary negative number we need to make sure that we pick a <Sign in to see all the formulas> whose minimum value will change if we have a different $a$ . We could choose <Sign in to see all the formulas>, <Sign in to see all the formulas>, etc. Since we only need <Sign in to see all the formulas> let's just choose <Sign in to see all the formulas>.

Now our guess for <Sign in to see all the formulas> is <Sign in to see all the formulas>. Now let's do the proof and see if this is a good guess for <Sign in to see all the formulas>.

We know that

<Sign in to see all the formulas>

Therefore,

<Sign in to see all the formulas>

So now let's assume that we are at an $x$ such that <Sign in to see all the formulas>. The $\min$ function gives us two cases: <Sign in to see all the formulas> or <Sign in to see all the formulas>. In either case we have <Sign in to see all the formulas>.

But remember that we made sure that <Sign in to see all the formulas>. We noted above that <Sign in to see all the formulas> means that <Sign in to see all the formulas> or <Sign in to see all the formulas>. In particular, <Sign in to see all the formulas> means that that <Sign in to see all the formulas> when <Sign in to see all the formulas>. Now let's do some algebra:

<Sign in to see all the formulas>

Now since <Sign in to see all the formulas> we also have that

<Sign in to see all the formulas>

Putting these together we have

<Sign in to see all the formulas>

Therefore,

<Sign in to see all the formulas>

The proof is now complete.