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Let $a, b$ be complex numbers. Use the definition of a limit directly (not just the properties of limits) to prove that $$ \lim_{z \to z_0}az + b = az_0 + b. $$

Sorry for the wrong notation, I do not know how to do it correctly. I just started this Complex Analysis class and I know how to prove this using the properties, but I feel like this is very arbitrary. Since we are taking the limit as $z \to z_0$, isn't this just obvious that the limit would be equal to substituting $z_0$ for $z$? Is this harder than I think it is.

Thanks for any help.

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    $\begingroup$ Do you know what the definition of limit is? It might seem "obvious," but the problem is asking you to do something very specific to show it, using a precise definition. $\endgroup$ Commented Jan 22, 2015 at 3:54

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Given $\epsilon \gt 0$, if there exists a $\delta \gt 0$ such that whenever $|z-z_0|\lt \delta$, we have $|f(z)-f(z_0)|=|a||z-z_0|$. Choose $\delta=\dfrac{\epsilon}{|a|}$ and we are done.

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