How does a sigmoid neuron act like a perceptron in this scenario?

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:

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.

• Please don't post text as pictures, rather copy paste referring to the source. Jul 25, 2022 at 9:17

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.