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Homework 3 (MATH 1199 Fall 2019)
1. Use the ϵ-δ definition of the limit to prove...
(a) lim
(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