Question

The Fresnel function S(x) = integral 0 to x sin (9/2 pi t^2) dt is important in the theory of Fourier Optics in Physics. Find the value of (Hint: remember L'Hospital's Rule.)

1. limit = 3/5
2. limit = 9/5 pi
3. limit = 3/10
4. limit = 3/10 pi
5. limit = 3/5 pi

Answer 1

`(3\pi)/(10)`

Explanation:

`S(x) = \int\limits_0^x \sin ((9)/(2) \pi t^2) dt\\\\\lim\limits_(x \to 0) (S(x))/(5x^3) =\lim\limits_(x \to 0) (\int\limits_0^x \sin ((9)/(2) \pi t^2) dt)/(5x^3)`

We will use L'Hospital Rule twice.

Firstly, to find the derivative of the integral, we will use Fundamental Theroem of Calculus.

`(d)/(dx) \lim\limits_(x \to 0) \int\limits_0^x \sin ((9)/(2) \pi t^2) dt= \sin ((9)/(2) \pi x^2)`

Hence,

`S(x) = \int\limits_0^x \sin ((9)/(2) \pi t^2) dt\\\\\lim\limits_(x \to 0) (S(x))/(5x^3) =\lim\limits_(x \to 0) (\int\limits_0^x \sin ((9)/(2) \pi t^2) dt)/(5x^3)=\lim\limits_(x \to 0) (\sin ((9)/(2) \pi x^2) )/(15x^2) =\lim\limits_(x \to 0) (9\pi x \cos((9)/(2) \pi x^2))/(30x) =\\\\=\lim\limits_(x \to 0) (9\pi \cos((9)/(2) \pi x^2))/(30)=(9\pi)/(30) =(3\pi)/(10)`