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Homework 3 (MATH 1199 Fall 2019)
1. Use the $\epsilon$-$\delta$ definition of the limit to prove...
(a) $\displaystyle\lim_{z \rightarrow 5} z+1 = 6$
(b) $\displaystyle\lim_{z \rightarrow 3i} \dfrac{z+1}{2} = \dfrac{3i+1}{2}$
(c) $\displaystyle\lim_{z \rightarrow \frac{1+i}{5}} \dfrac{5z-i}{7} = \dfrac{1}{7}$

2. Show that $\displaystyle\lim_{z \rightarrow 0} \dfrac{z}{|z|}$ does not exist by demonstrating two paths to zero that yield different results.

3. Use the limit theorem for $\infty$ to explain the following limits. Write $\epsilon$-$\delta$ proofs as-needed.
(a) $\displaystyle\lim_{z \rightarrow \infty} \dfrac{1}{z} = 0$
(b) $\displaystyle\lim_{z \rightarrow \infty} \dfrac{4z^2}{(z-1)^2} = 4$
(c) $\displaystyle\lim_{z \rightarrow \infty} \dfrac{z^2+1}{z-1} = \infty$