University:
University of California, Berkeley
Course:
MATH 51 | Calculus I
Academic year:

2022
Views:

360

Pages:

1
Author:

Clint W.

Let’s review limits from single variable calculus. Definition: S uppose f (x) i s d efined on an op en in terval ab out c, ex cept po ssibly at c it self. If f (x) is arbitrarily close to L as we like for all x sufficiently cl ose to c, we sa y th at f approaches the limit L as x approaches c, and write limx→c f (x) = L, which is read “the limit of f (x) as x approaches c is L.” Definition: We u se t he notation lim(x,y)→(a,b) f (x, y) = L to indicate that the values of f (x, y) approach the number L as the point (x, y) approaches the point (a, b) along any path that stays within the domain of f . Exercise 1. Find (a) lim(x,y)→(2,−3) (x3 − 4xy 2 + 5y − 7) 2 2 (b) lim(x,y)→(3,4) √x −y 2 2 x +y Two-Path Rule: If two different paths to a point P (a, b) produce two different limiting values for f , then lim(x,y)→(a,b) f (x, y) does not exist. Exercise 2. Show that lim(x,y)→(0,0) x2 −y 2 x2 +y 2 does not exist. (Swok Sec 16.2 Ex 2) Exercise 3. Show that lim(x,y)→(0,0) x2 −y 2 x2 +y 2 does not exist. (Swok Sec 16.2 Ex 3) Exercise 4. Show that lim(x,y)→(0,0) x2 y x4 +y 2 does not exist. (Swok Sec 16.2 Ex 4) Class Exercise 1. Find the limit, if it exists, or show that the limit does not exist. (#6-18 even) 2 (a) lim(x,y)→(1,−1) e−xy cos(x + y) (b) lim(x,y)→(1,0) ln( x1+y 2 +xy ) 5y 4 cos2 x x4 +y 4 x4 −y 4 lim(x,y)→(0,0) x2 +y2 (c) lim(x,y)→(0,0) (e) xy−y (x−1)2 +y 2 2 sin y lim(x,y)→(0,0) xx2 +2y (g) 2 (d) lim(x,y)→(1,0) (f) 2 lim(x,y)→(0,0) xy 4 x2 +y 8 Definition: The notation lim(x,y,z)→(a,b,c) f (x, y, z) = L means that the values of f (x, y, z) approach the number L as the point (x, y, z) approaches the point (a, b, c) along any path in the domain of f . Class Exercise 2. Find the limit, if it exists, or show that the limit does not exist (#20,22) (a) lim(x,y,z)→(0,0,0) x2xy+yz (b) lim(x,y,z)→(0,0,0) x2 +4yyz2 +9z2 +y 2 +z 2 Definition: A function f of two variables is called continuous at (a, b) if lim(x,y)→(a,b) f (x, y) = f (a, b). We say that f is continuous on D if f is continuous at every point (a, b) in D. Exercise 5. If h(x, y) = ex (Swok Sec 16.2 Ex 7) 2 +5xy+y 2 Exercise 6. Show that f (x, y) = , show that h is continuous at every pair (a, b). 2xy x2 +y 2 0, if(x, y) ̸= (0, 0) is continuous at every point except if(x, y) = (0, 0) at the origin. (Hass Sec 14.2 Ex 5) Exercise 7. Determine the points at which the following functions are continuous. (a) h(x, y) = ln(x2 + y 2 + 4) (b) h(x, y) = ex/y (Briggs Sec 12.3 Ex 5) Class Exercise 3. the function is continuous. (#30-36 even) √ Determine the set of pointsexat+ewhich y (a) F (x, y) = cos 1 + x − y (b) H(x, y) = exy −1 p (c) G(x, y) = tan−1 ((x + y)−2 ) (d) f (x, y, z) = y − x2 ln z Homework: 5-29 (every 4th), 37-49 (every 4th)