This course doesn't open with new mathematics — it opens by asking a practical question about mathematics you already know how to do exactly: what happens the moment you can't do it exactly anymore? Solving Ax=b by hand, factoring a characteristic polynomial, integrating a function symbolically — every one of these has a point where the problem gets too large, too irregular, or too expensive for the exact method to finish. Numerical analysis is the discipline of what to do next.
This chapter is a fast, deliberate pass over six tools this course leans on immediately: Taylor's theorem (the source of nearly every error bound that follows), the limit of a sequence (what "converges" means before this course asks how fast), Gaussian elimination (the exact procedure this course's own linear-systems chapters make numerically honest), eigenvalues and eigenvectors (computed exactly here, computed without factoring a polynomial starting in chapter 10), vector norms (how "how big is the error" gets a precise number), and Euler's method (the simplest possible numerical ODE solver, and the floor this course spends a whole chapter building above).
Nothing here is new material, and every practice question is pulled straight from the course that first taught it. By the end of this chapter, every tool this course leans on is back on the bench — the next chapter starts building with them.