What is sin(x)/x for x =0?

I'm going to show the answer to this question in two different ways. - The first is perhaps more obvious but the second is much more elegant.Taylor series expansion: Using Taylor expansion (or your trusty A level formula sheet) you can show that sin(x) = x - x^3/3! + x^5/5! + Re ( (-i)^n * x^n / n! )Thus dividing through by x:sin(x)/x = 1 - x^2/3! + x^4/5! +...if we then replace x by 0:sin(0)/0 = 1 - 0 + 0 +... where ... here is all 0.thus sin(0) / 0 = 1. The other much faster way of doing this is using l'Hopital's rule which states that for a limit lim (f(x)/g(x)) = lim (f'(x) / g'(x)) for the same limit. Thus lim[x-> 0] (sin(x) / x) = lim[x->0] (cos x / 1) = 1.

VP
Answered by Vandan P. Further Mathematics tutor

40653 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

For a homogeneous second order differential equation, why does a complex conjugate pair solution (m+in and m-in) to the auxiliary equation result in the complementary function y(x)=e^(mx)(Acos(nx)+Bisin(nx)), where i represents √(-1).


Solve the second order differential equation d^2y/dx^2 - 4dy/dx + 5y = 15cos(x), given that when x = 0, y = 1 and when x = 0, dy/dx = 0


Find the general solution of the differential equation d^2y/dx^2 - 5*dy/dx + 4y = 2x


Find the area of the surface generated when the curve with equation y=cosh(x) is rotated through 2 pi radians about the x axis, with 2<=x<=6


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences