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 [itex]\epsilon[/math] definition of a limit.

2. The attempt at a solution

Well, I can't do $\frac{1}{a} - \infty < \epsilon$ can I? Otherwise it's just obvious that it'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 :

$\forall M>0, \exists δ>0 \space | \space 0<|x-c|<δ \Rightarrow f(x) > M$

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.