How do I substitute $w=-1$ and $z=\infty$ in my equation?

Question

I am given $w=-1$ and $z=\infty$.

I also know that $\frac{1}{\infty} =0$.

How do I substitute the value of $z$ and $w$ in $$w=\dfrac{az+b}{cz+d}$$ without complicating the equation?

Answer 1

I assume you mean how do I let $z$ approach infinity and $w=-1$ to get a simplified equation. Factor the $z$ on the top and bottom to get:

$$w=\frac{a+\frac{b}{z}}{c+\frac{d}{z}}$$

Now let $z \to \infty$

To get

$$w=\frac{a}{c}=-1$$

However Like Andre said, if we have $a=c=0$, then the equation is independent of $a$ and $c$, and it reduces to:

$$w=\frac{b}{d}=-1$$