# Basic limit question

1. Apr 14, 2010

### complexnumber

1. The problem statement, all variables and given/known data

\begin{align*} f(t) = \lim_{k \to \infty} f_k(t) = \lim_{k \to \infty} \frac{1 - kt^2}{1 + kt^2} = \lim_{k \to \infty} \frac{\frac{1}{k} - t^2}{\frac{1}{k} + t^2} = \frac{0 - t^2}{0 + t^2} = - \frac{t^2}{t^2} \end{align*}

What is the value of limit function $$f$$ when $$t = 0$$? Is it $$0$$ or $$-1$$ or undefined? What is the reasoning behind it?

Does anyone know any good websites or books to catch up on these material?

2. Relevant equations

3. The attempt at a solution

2. Apr 14, 2010

### HallsofIvy

None of the above!

If $t\ne 0$ then the limit is -1, obviously.

If t= 0, go back to the original formula: if t= 0, then
$$\frac{1- kt}{1+ kt}= \frac{1- 0}{1+ 0}= \frac{1}{1}= 1$$
which is independent of k. The limit, if t= 0, is 1.

3. Apr 14, 2010

### complexnumber

When should I use the original formula first and when should I take the limit first?