This is problem from here, but I didn't receive any answers $\lim_{x\to \infty}e^{-x^2}\int_x^{x+1/x}e^{t^2}dt$.
I need to differentiate $\int_x^{x+1/x}e^{t^2}dt$ and I got $(1- 1/x^2) e^{(x+1/x)^2}-e^{x^2}$
Am I right? I am asking because Spivak Calculus 1994 in chapter 18 problem 31, Calculus 1994 got another answer. He got $e^{(x+1/x)^2}-e^{x^2}$
I think he is missing derivative of $x+1/x$
Assume we have a function $F(x)$ such that:$$F'(x)=e^{x^2}$$therefore$$\int_{x}^{x+{1\over x}}e^{t^2}dt=F(x+\dfrac{1}{x})-F(x)$$by differentiating we get:$$\dfrac{d}{dx}\int_{x}^{x+{1\over x}}e^{t^2}dt=(1-\dfrac{1}{x^2})F'(x+\dfrac{1}{x})-F'(x)=(1-\dfrac{1}{x^2})e^{(x+\dfrac{1}{x})^2}-e^{x^2}$$