Question

can somebody tell me what this is asking?

Answer 1

The function for the tangent line is often called the first order approximation of 𝑓(𝑥) at 𝑥=0, because it approximates the function with a straight line. The Taylor polynomial of degree 2 is often called the second order approximation of 𝑓(𝑥) at 𝑥=0, because it approximates the function with a parabola. You can find higher order Taylor polynomials as well, to get third order approximations and so on. Show the following for your answers from (a) & (b):
i. lim𝑥→0𝑓 𝑥 −𝐿 𝑥 𝑥=0 but
ii. lim𝑥→0𝑓 𝑥 −𝐿 𝑥 𝑥2≠0 (This means 𝐿(𝑥) is only a first order approximation, not second order)
iii. lim𝑥→0𝑓 𝑥 −𝑄 𝑥 𝑥2=0 (This means 𝑄(𝑥) is a second order approximation.)