Write the tangent line in point-slope form: $y-y_1 = m(x-x_1)$

Find the point by evaluating $g(0)$. Careful! Notice the denominator.

Though the denominator 'cancels out', there remains a hole in the graph at $z = 0$. Since there is no point, there is no line, and consequently no tangent line.

Step 2 would be to use $g^\prime(z)$ to find the slope at $z = 0$. However, since the tangent line does not exist, there is no reason to find its slope!