Prove function is continuous using epsilon delta

Hello,

The function given is f(z)=(z^2+3)/(z+1) and the domain is the complex plane C\{-1}.
Now let epsilon e>0 be given, so that
|f(z)-f(zo)|=|(z-zo)(z^2-zo^2+zzo-3)/(…
now I can set delta d=(z+1)(zo+1)e/(2(z^2+zo^2+zzo-3)) and it will work but I can't show that
the denominator of delta is not zero..or even factor.
Any help will be appreciated.

Thank you

Note that |f(z) - f(z₀)|
= |[(z^2 + 3)(z₀ + 1) - (z₀^2 + 3)(z + 1)] / [(z₀ + 1)(z + 1)]|
= |[(z₀ z^2 + z^2 + 3z₀ + 3) - (z z₀^2 + z₀^2 + 3z + 3)] / [(z₀ + 1)(z + 1)]|.

Expand the numerator into powers of (z - z₀):
|f(z) - f(z₀)|
= |[(z₀ z^2 - z z₀^2) + (z^2 - z₀^2) - 3(z - z₀)] / [(z₀ + 1)(z + 1)]|
= |z₀ z (z - z₀) + (z - z₀)(z + z₀) - 3(z - z₀)| / |(z₀ + 1)(z + 1)|
= |z₀ ((z - z₀) + z₀) (z - z₀) + (z - z₀)((z - z₀) + 2z₀) - 3(z - z₀)| / |(z₀ + 1)(z + 1)|
= |z₀ (z - z₀)² + z₀² (z - z₀) + (z - z₀)² + 2z₀ (z - z₀) - 3(z - z₀)| / |(z₀ + 1)(z + 1)|

Now, apply the triangle inequality:
|f(z) - f(z₀)|
≤ [|z₀| |z - z₀|² + |z₀|² |z - z₀| + |z - z₀|² + 2|z₀| |z - z₀| + 3 |z - z₀|] / |(z₀ + 1)(z + 1)|
< [|z₀| |z - z₀| + |z₀|² |z - z₀| + |z - z₀| + 2|z₀| |z - z₀| + 3 |z - z₀|] / |(z₀ + 1)(z + 1)|,
assuming that |z - z₀| < 1
= [|z₀|² + 3|z₀| + 4] |z - z₀| / (|z₀ + 1| |z + 1|).
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Now, we need to bound |z + 1|.

Assume even further that |z - z₀| < |z₀ - (-1)|/2 to avoid z = -1 completely
==> -|z₀ + 1|/2 < z - z₀ < |z₀ + 1|/2
==> (z₀ + 1) - |z₀ + 1|/2 < z + 1 < (z₀ + 1) + |z₀ + 1|/2.

If (z₀ + 1) > 0, then we take |z + 1| > (z₀ + 1) - |z₀ + 1|/2 = |z₀ + 1|/2.
If (z₀ + 1) < 0, then we take |z + 1| > -(z₀ + 1) + |z₀ + 1|/2 = 3|z₀ + 1|/2.

At any rate, we have |z + 1| > 3|z₀ + 1|/2.

So, [|z₀|² + 3|z₀| + 4] |z - z₀| / (|z₀ + 1| |z + 1|)
< [|z₀|² + 3|z₀| + 4] (|z - z₀| / |z₀ + 1|) * (2/(3|z₀ + 1|))
< [|z₀|² + 3|z₀| + 4] (|z - z₀| / |z₀ + 1|²)
= M |z - z₀|, where M = [|z₀|² + 3|z₀| + 4] / |z₀ + 1|², independent of z.
--------------------------------------…
So given ε > 0, let δ = ε/M.
Then, 0 < |z - z₀| < δ ==> |f(z) - f(z₀)| < M |z - z₀| < M(ε/M) = ε, as required.

I hope this helps!