Solving Basic Limit Questions: When t=0, is it 0, -1, or Undefined?

[complexnumber]

Homework Statement

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

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?

Homework Equations

The Attempt at a Solution

 

[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.

 

[complexnumber]

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.

Thanks for your reply. I have one question about getting to the solution.

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