Question about cost function for logistic regression

Question

During a lecture on logistic regression in coursera, there was a question about the cost function as shown in the following images:

I don't understand why the third unselected option is true?

Answer 1

For logistic regression, $h_\theta(x)=\frac{1}{1 + e^{-\theta^{T}x}}$ is the sigmoidal function. So, $-\log(1 - h_\theta(x))$ = $-\log{\frac{1}{1 + e^{\theta^{T}{x}}}}$. So, if $h_\theta(x)$ tends to $0$ then $e^{\theta^{T}x} = 0$, So, the cost should be $0$ should not be $-\infty$, which makes sense since the cost should be minimum for correct prediction where $h_\theta(x)$ matches the actual $y$ value.