Calculus: Early Transcendentals, Chapter 4, 4.4, Section 4.4, Problem 14

lim_(x->0) x^2/(1-cos(x))
The function f(x) = x^2/(1-cosx) is undefined at x=0. So to compute its limit as x
approaches zero, apply the L'Hospital's Rule. Take the derivative of the numerator and denominator.
=lim_(x->0) ((x^2)')/((1-cos(x))')= lim_(x->0) (2x)/sin(x)
The resulting function (2x)/sin(x) is still undefined at x=0. So, take the derivative of the numerator and denominator again.
=lim_(x->0) ((2x)')/((sin(x))')= lim_(x->0) 2/cos(x)
Now that the resulting function is defined at x=0, proceed to plug-in this value of x.
= 2/cos(0) = 2/1=2

Therefore, lim_(x->0)x^2/(1-cos(x))=2.