Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Quiz 2 => Topic started by: Pengyun Li on October 01, 2020, 07:08:28 PM

$\textbf{Question}$: Find the limit of each function at the given point, or explain why it does not exist: $f(z) = (z2)logz2$ at $z_0 = 2$.
$\textbf{Answer}$: Since $z_0= 2$, let $z' = z2$.
$$\lim_{z\to z_0} f(z) = \lim_{z' \to 0} f(z) =z'\log z'$$
$$=z' (\ln z' +i\cdot 0) = \frac{ln z'}{\frac{1}{z'}}$$
By L'HÃ´pital's Rule, $$= \frac{\frac{1}{z'}}{\frac{1}{(z')^2}}=\lim_{z'\to 0} (z') = 0$$.
Therefore, the limit of the function is 0 at $z_0=2$.