Limit concept under Complex analysis

Question

Prove that $$ \lim_{z\to i} \dfrac{3z^4-2z^3+8z^2-2z+5}{z-i} = 4+4i $$

Answer 1

Hint: The number $i$ is a root of both the denominator and the numerator. Since the numerator is a polynomial with real coefficients, it follows that $-i$ is also a root of the polynomial, which means that you can factor out $z^2+1$ on the numerator, which should be easier than factoring out $z-i$.

Answer 2

The numerator contains $z-i$ as a factor so it will be of the form ($z-i$)$f(z)$ where $f(z)$ is some polynomial in $z$ which you can evaluate at i?