Calculating the Value of a complex limit
By Joseph Russell •
I am given some limits that exist, I'm supposed to find their values. Seems really simple however I am struggling.
Find the value of $\displaystyle \lim_{z\to\\i}\frac{z^4 - 1}{z-i} $.
My approach was to paramaterize $z$ to $it$ and transform the limit to something like:
$$\lim_{t\to\\1}\frac{(it)^4 - 1}{it-i} $$
Could anyone lend a hand as to how to solve something like this?
$\endgroup$ 21 Answer
$\begingroup$(1)L'Hospital directly
$$\lim_{z\to i}\frac{z^4-1}{z-i}=\lim_{z\to i}4z^3=-4i$$
(2) Factoring:
$$z^4-1=(z^2-1)(z-i)(z+i)\Longrightarrow \lim_{z\to i}\frac{z^4-1}{z-i}=\lim_{z\to i}(z^2-1)(z+i)=-2(2i)=-4i$$
$\endgroup$ 4More in general
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