Prove [itex]\lim_{a\to 0}\frac{1}{a} = \infty[/itex]

[operationsres]

Homework Statement

Prove \lim_{a\to 0^+}\frac{1}{a} = +\infty under the \epsilon[/math] definition of a limit.<br /> <br /> <b>2. The attempt at a solution</b><br /> <br /> Well, I can&#039;t do \frac{1}{a} - \infty &amp;lt; \epsilon can I? Otherwise it&#039;s just obvious that it&#039;s infinity ..

 

[STEMucator]

For this particular problem you need to alter your definition abit since |f(x) - ∞| < ε translates into a useless statement.

You want to use this definition :

\forall M&gt;0, \exists δ&gt;0 \space | \space 0&lt;|x-c|&lt;δ \Rightarrow f(x) &gt; M

What this definition essentially means is that we can find a delta such that the function grows without bound.

Start by massaging the expression f(x) > M into a suitable form |x-c| < δ which will give you a δ which MIGHT work.

Then take that δ and show that it implies f(x) > M.