14.2 Limits and continuity in several variables Recall that on the real number line, a number X is close to a ifqthe distance |x-a| is small. in the plane, a point (x,y) is close to another point P = (a,b) if the distance d ((x; y) ; (a; b)) = (x ¡ a) + (y ¡ b) between them is small 2 2 Note that if we take all the points that are a distance of less than r from P = (a,b), this is a disk D(P,r) centered At P that does not include its boundary Now assume that f(x,y) is defined near p but not necessarily at P itself. in other canexcludesp y. words, f(x,y) is defined for all (x,y) In some punctured disk D*(P,r) with r > 0. We say that f(x,y) approaches the limit L as (x,y) approaches p = (a,b) if |f(x,y) - L| becomes arbitrarily small for (x,y) sufficiently close to P = (a,b) ay lim f (x; y) = opendiskDCP.rs (x;y)!P lim (x;y)!(a;b) f (x; y) = L In a one-variable limit, we require that F(x) tend to l as x approaches a from two directions - the left and the right. in a multivariable limit, f(x,y) must tend to L as (x,y) approaches p from infinitely many different directions Theorem 1: i. Sum law: lim (f (x; y) + g (x; y)) = lim f (x; y) + lim g (x; y) (x;y)!p (x;y)!p (x;y)!p ii. Constant multiple law: lim kf (x; y) = k lim f (x; y) (x;y)!p (x;y)!p iii. Product law: lim f (x; y) g (x; y) = (x;y)!p µ lim f (x; y) (x;y)!p ¶µ iv. Quotient law: lim (x;y)!p lim(x;y)!p f (x; y) f (x; y) = g (x; y) lim(x;y)!p g (x; y) lim g (x; y) (x;y)!p ¶ As in the single-variable case, we say that F is continuous at P = (a,b) if f(x,y) approaches the value of the function f(a,b) as (x,y) —> (a,b) When we apply them to f(x,y) = x and g(x,y) = y, which are continuous, we find that the power functions are continuous for all whole numbers m,n and that all polynomials are continuous Composition is another important way to build functions. If F is a function of two variables and G(u) a function of one variable, then the composite function G * f is the function of two variables given by g(f(x,y)) to prove a limit does not exist, we only need to find two paths that yield different limits. However, to prove that a limit does exist at a point, it is not enough just to consider the limit along a set of paths approaching the point