Answers.Body: <blockquote>
  <p>When c approaches infinity, wouldn’t make the sigmoid function always
  output a value close to 1 whereas a perceptron can output 0 or 1.</p>
  </blockquote>
  <p>This is true only when the original value was positive.</p>
  <p><span class="math-container">$e^{-cx}$</span> coverges to <span class="math-container">$0$</span> (and <span class="math-container">$\frac{1}{1 + e^{-cx}}$</span> converges to <span class="math-container">$1$</span>) as <span class="math-container">$c$</span> approaches infinity only when <span class="math-container">$x$</span> is positive. If x is negative, <span class="math-container">$e^{-cx}$</span> diverges to infinity and thus <span class="math-container">$\frac{1}{1 + e^{-cx}}$</span> converges to <span class="math-container">$0$</span>.</p>
  <p>Thus, the sigmoid function behaves like a step function as <span class="math-container">$c$</span> approaches infinity.</p>
  
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Answers.CreationDate: 2022-07-25T08:51:16.150Z
Answers.LastActivityDate: 2022-07-25T08:51:16.150Z
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Body: <p>I have been reading Michael Nielsen’s book online on his website at <a href="http://neuralnetworksanddeeplearning.com/chap1.html" rel="nofollow noreferrer">http://neuralnetworksanddeeplearning.com/chap1.html</a>. I am struggling to understand the second exercise:
  <a href="https://i.sstatic.net/GQNVi.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GQNVi.jpg" alt="enter image description here" /></a></p>
  <p>When c approaches infinity, wouldn’t make the sigmoid function always output a value close to 1 whereas a perceptron can output 0 or 1.</p>
  <p>Let me know if I am missing something or maybe if someone can rephrase the question in a clearer way.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
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Comments.Text: Please don't post text as pictures, rather copy paste referring to the source.
Comments.UserDisplayName: Martino
Tags: <neural-networks><perceptron><sigmoid>
Title: How does a sigmoid neuron act like a perceptron in this scenario?
UserDisplayName: QuantNoob
ThreadText: # How does a sigmoid neuron act like a perceptron in this scenario?
  
  (Asked by: username_0 on 2022-07-24)
  
  I have been reading Michael Nielsen’s book online on his website at <http://neuralnetworksanddeeplearning.com/chap1.html>. I am struggling to understand the second exercise:
  [![enter image description here](https://i.sstatic.net/GQNVi.jpg)](https://i.sstatic.net/GQNVi.jpg)
  
  When c approaches infinity, wouldn’t make the sigmoid function always output a value close to 1 whereas a perceptron can output 0 or 1.
  
  Let me know if I am missing something or maybe if someone can rephrase the question in a clearer way.
  
  **Tags:** neural-networks, perceptron, sigmoid
  
  > **Comments:**
  > **username_1**: Please don't post text as pictures, rather copy paste referring to the source.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2022-07-25. Score: 1
  
  > When c approaches infinity, wouldn’t make the sigmoid function always
  > output a value close to 1 whereas a perceptron can output 0 or 1.
  
  This is true only when the original value was positive.
  
  $e^{-cx}$ coverges to $0$ (and $\frac{1}{1 + e^{-cx}}$ converges to $1$) as $c$ approaches infinity only when $x$ is positive. If x is negative, $e^{-cx}$ diverges to infinity and thus $\frac{1}{1 + e^{-cx}}$ converges to $0$.
  
  Thus, the sigmoid function behaves like a step function as $c$ approaches infinity.
  
  
  ---
  
  