Please help please show the work as well. i know what the answer is, just not sure how to get there. 25 pts.

D) 1/e but you probably know already

Step-by-step explanation:

The Fundamental Theorem of Calculus part 1:

A similar argument where a= -infinity and b=1 gives you d/dx f(x) = g(x) for all x in [-infinity, 1]. Therefore it holds for all x in [-infinity, infinity]

Also, d/dx (x+1) = 1

Also also, f(1)=0 because the bounds become equal

Thus the limit is an indeterminate form of 0/0; this suggests using L'Hopital's Rule

By definition of continuity and the fact that the above g(x) converges at 1, we have

*there's a proof on proofwiki.org that's too long to fit here, but you probably don't need it

**it's 3am, maybe I'll prove it in the morning if I have time lol. Anyway if want to do it yourself, you can use the limit definition of congruence, drag the limit symbol to the exponent -x^2, etc. Or just apply the theorem without checking for continuity. Tell me if you need more detail (or less rambling)

mnkjn

step-by-step explanation: