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

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Lesson Parent:

Problem:

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

Answer:

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

Solution:

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 positive 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>. Since $a$ is any positive arbitrary 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>.

We know that

Therefore,