MATH 151 Engineering Mathematics I Fall 2018, WEEK 3 JoungDong Kim Week 3 Section 2.3, 2.5, 2.6, Calculating Limits Using the Limit Laws, Continuity, Limits at Infinity; Horizontal Asymptotes. Section 2.3. Calculating limits using the limit laws Limit Laws Suppose that c is a constant and the limits exist. Then limf(x) and lim g(x) 1. 2. 3. 4. 5. 6. 7. 8. lim[f(x)+g(x)] = limf(x)+ lim g(x) lim[f(x) g(x)] = limf(x) lim g(x) lim[cf(x)] = clim f(x) lim[f(x) g(x)] = limf(x) lim g(x) f(x) limf(x) lim g(x) = lim g(x) if limg(x) 0 lim [f(x)]n = [limf(x)] n lim c = c lim x = a 1
9. 10. 11. lim lim xn = a n lim n x = n a n f(x) = n f(a) where n is a positive integer. (If n is even, we assume that lim f(x) > 0). Ex.1) If it is known that limf(x) = 3, find x 2 f(x)+6 a) lim x 2 f(x)+2x+11 b) lim x 2 (f(x)) 2 2
How to find limits algebraically lim f(x) Try plugging a into the function: 1. If you get a real number, that is your answer (unless you are dealing with a piecewise function) 2. If you get 0 (indeterminate form), algebraically manipulate (usually factor), cancel, and 0 plug in a again. none zero number 3. If you get then your answer could be + or. 0 Ex.2) Find limit a) lim x 2 x 3 +2x 2 1. 5 3x b) lim x 2 (x2 +x+1) 5 c) lim x 1 x 4 +x 2 6 x 4 +2x+3 3
x 2 x 12 d) lim x 3 x+3 e) lim x 9 x 2 81 x 3 x 3 f) lim x 3 x2 +7 4 g) lim x 2 x 4 (x 2) 2 4
h) lim h 0 (4+h) 1 4 1 h 1 t i) lim t 1 t 2 1, 2t Ex.3) Show that lim x 0 x = 0 5
x Ex.4) Prove that lim x 0 x does not exist. Ex.5) Let f(x) = x2 +3x, find lim x+3 f(x). x 3 6
The Squeeze Theorem If f(x) g(x) h(x) for all x in open interval that contains a (except possibly at a) and limf(x) = lim h(x) = L then lim g(x) = L Ex.6) Show that lim x 0 xsin 1 x = 0. 7
Section 2.5 Continuity Definition A function f is continuous at a number a if limf(x) = f(a) Note Definition implicitly requires three things if f is continuous at a; 1. f(a) is defined (that is, a is in the domain of f and has a function value at a) ( ) 2. lim f(x) exists. 3. lim f(x) = f(a) lim +f(x) = lim f(x) 8
In each case the graph cannot be drawn without lifting the pen from the paper, because a hole or break or jump occurs in the graph. Definition. The kind of discontinuity illustrated first and three is called removable becuase we could remove the discontinuity by redefining f at 2. The discontinuity in second figure is called an infinite discontinuity. The discontinuities in the last figure are called jump discontinuities because the function jumps from one value to another. Definition. A function f is continuous from the right at x = a if lim +f(x) = f(a) and if f is continuous from the left at x = a if lim f(x) = f(a) 9
Theorem. (a) Any polynomial is continuous everywhere; that is, it is continuous on R = (, ). (b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain. Ex.7) Show that f(x) = x 2 3x+2 is continuous at x = 2. Ex.8) From the accompaning figure, state the numbers at which f is discontinuous. For each of the numbers stated, state whether f is continuous from the right, or from the left, or neither. 10
Ex.9) Explain why the following functions are not continuous at the indicated values of x. a) f(x) = 1 (1 x) 2, x = 1 2x+1 if x 0 b) f(x) = 3x if x > 0, x = 0 x 2 2x 8 if x 4 c) f(x) = x 4 3 if x = 4, x = 4 11
x 2 c 2 if x < 4 Ex.10) If g(x) = cx+20 if x 4. For what value(s) of c make(s) g(x) continuous? 12
Intermediate Value Theorem Suppose that f is continuous on the closed interval [a,b] and let N be any number strictly between f(a) and f(b). Then there exists a number c in (a,b) such that f(c) = N. Ex.11) Show there is a root of the equation 4x 3 6x 2 +3x 2 = 0 between 1 and 2. Ex.12) If g(x) = x 5 2x 3 +x 2 +2, show there a number c so that g(c) = 1 in [ 2, 1]. 13
Section 2.6 Limits at Infinity; Horizontal Asymptotes Definition. Let f be a function defined on some interval (a, ). Then lim f(x) = L means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Definition. Horizontal Asymptote The line y = L is called a horizontal asymptote of the curve y = f(x) if either lim f(x) = L or lim f(x) = L. x Ex.13) Find limit. a) lim arctanx b) lim x arctanx 14
Limit of Power Function at Infinity If p is a positive real number, lim x ± 1 x p = 0. Finding limits at infinity for a rational function, f(x): Look for the highest degree of x: 1. If it is in the denominator, then lim f(x) = 0. x ± 2. If it is in the numberator, then lim f(x) = ±. x ± 3. If the degree of the polynomial in the numberator and denominator is the same then f(x) = ratio of the leading coefficients. lim x ± Ex.14) Find the limits: a) lim x b) lim x x c) lim (x x 2 ) d) lim x (x x3 ) 15
1 e) lim x 1 f) lim x x 1 g) lim x 4 h) lim 7x 3 +4x 2x 3 x 2 +3 i) lim t t 4 t 2 +1 t 5 +t 3 t x 4 +2x+3 j) lim x x(x 2 1) k) lim 1+4x 2 4+x l) lim x 1+4x 2 4+x m) lim x x2 +4x 4x+1 16
n) lim (0.3) x o) lim x (0.3) x p) lim x 2 + ( ) x 1 2 x 4 q) lim x 2 ( ) x 1 2 x 4 r) lim 2 x 1 e x 1 s) lim 2 x 1 +e x 1 t) lim e x e 3x e 3x +e 3x 17
u) lim ( x 2 +3x+1 x) v) lim x (x+ x 2 +2x) 18
Theorem. If a > 1, the function f(x) = log a x is one-to-one, continuous, increasing function with domain (0, ) and range R. If x, y > 0, then 1. log a (xy) = log a x+log a y ( ) x 2. log a = log y a x log a y 3. log a (x y ) = ylog a x 4. log a a = 1 5. a log a x = x 6. change of base: log a b = log cb log c a or log ab = 1 log b a 7. log a 1 = 0 Ex.15) Find the limit; a) lim (ln(3x 2 2x+5) ln(2x 2 +4x)) 19
b) lim (ln(3x 2 ) ln(6x 4 3x+1)) Ex.16) Find the limits; a) lim arctan(e x ) b) lim arctan(lnx) c) lim x 0 +arctan(lnx) 20
Finding the Vertical Asymptote and Horizontal Asymptote. 1. Vertical asymptote: undefined point but if it could be cancelled, it is not vertical asymptote but hole. 2. Horizontal asymptote: use infinite limit x and x. Ex.17) Find the equation of all vertical and horizontal asymptotes. a) f(x) = x+3 x 2 +7x+12 b) f(x) = x x2 +1 21