Chapter 1 Summary Math 1231: Single-Variable Calculus

Roots or Zeros: Definition Given a function f(x), we call x = a is a ROOT or ZERO of f(x) if f(a) = 0. Given a equation f(x)=0, we call x = a is a root or zero or SOLUTION of the equation if f(a) = 0. Example: x = π/2 is a root of cos(x); Example: x = 1 is a root of x 2 + x = x + 1.

Limit lim x->a f(x) = L exists if, for ANY sequence of x values that approach a, f(x) always becomes sufficiently close to L. Example: lim x->2 x 2 = 4 means, for any sequence of x values, x = 2.1, 2.01, 2.001, , … x = 1.9, 1.99, 1.999, , … x = 2.2, 1.8, 2.02, 1.98, 2.002, 1.998, … x 2 always sufficiently approaches 4.

When Does Limit Not Exist? If f(x) approaches two different values when x approaches a along two different sequences, then f(x) does not have a limit. Example: f(x) = sin(1/x) does not have a limit when x approaches 0, because by taking x to be x = 1/π, 1/2π, 1/3π, …, f(x)  0 x = 1/0.5π, 1/2.5π, 1/4.5π, …, f(x)  1

More Examples For any point x = a, lim x-> a f(x) does not exist. So f(x) is discontinuous everywhere. lim x-> a f(x) does not exist for all the points except x = 0. Furthermore, f(x) is continuous only at x = 0.

Continuity 24 f(x) = f 1 (x)f(x) = f 2 (x)f(x) = f 3 (x)

Continuity f(x) is continuous at a if lim x->a f(x) = f(a) Continuity implies we can switch the order of lim and f Example: Suppose that lim x->1 g(x) = π, find lim x->1 cos(g(x)).

Miscellaneous sine and cosine functions are bounded by -1 and 1.

Miscellaneous I.V.T. is only valid when f(x) is continuous.