Question

Let P be a point on the graph of y=e−x2/16 with NONZERO x-coordinate a. The normal line to the graph through P will have y-intercept b. (Note: The normal line at P is perpendicular to the tangent line at P.)

Answer 1

Yes. Just looking at the figure should confirm your conclusion: as P approaches the y-axis, the slope of the the normal becomes vertical and b is completely undetermined.

There is ambiguity in where the "3" goes in the exponent. You need more parentheses or curly brackets:
e^{-x^2/3} or e^{-x^{2/3}}
when expressed as LaTeX code, becomes
e−x2/3 or e−x2/3