Question

Let f(z) = z/4i. Use the ε − δ definition of a limit to show that lim z→2 f(z) = 1/2i.

          Let f(z) = z/4i. Use the ε − δ definition of a limit to show that lim z→2 f(z) = 1/2i.
        

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Let f(z) = z/4i. Use the ε − δ definition of a limit to show that lim z→2 f(z) = 1/2i.
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Transcript

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00:01 In this problem, we're thinking about the limit in the complex plane as something goes to infinity.
00:06 And so if we say that the limit of z approaches z -not of f -of -z is equal to infinity, what this really means is that for any m greater than zero, there exists some epsilon greater than zero, such that if the distance between z and z -not is less than epsilon, this implies that the absolute value of f of z is greater than m.
00:33 Now, what we want to do is we want to show that this implies that the limit as z goes to z -not of 1 over f of z is equal to 0.
00:47 And the way that we can do that is we can write this as g of z.
00:51 So g of z is equal to 1 over f of z.
00:54 And now let's write down the limit, the finite limit here.
00:59 So what that really says is that for every delta greater than zero, there exists...
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