Continuity Worksheet Key

Calculus AB Continuity Worksheet Name Key State whether or not each of the following functions is continuous. If not, state where the discontinuity occurs and whether or not it is removable. Is the discontinuity is an asymptote, a hole, or a jump? If it is an asymptote, what is its equation? 1) f(x) = x/(x ^ 2 + 1) continuous 2) f(x) = x/(2x ^ 2 - x - 1) = x/((2x + 1)(x - i)) not continuous ot x = - 1/2 $ x = 1 VA x = - 1/2 x = 1 (non-remove.ble) 3) f(x) = (2x + 3)/(x ^ 2 - x - 6) = (2x + 3)/((x - 3)(x + 2)) not continuous at x = 3 and x = - 2 VA x = 3 x = - 2 () 4) f(x) = (x - 4)/(x ^ 2 - 16) = (x - 4)/((x + 4)(x - 4)) = 1/(x + 4) anat continuas at x=-4 A x = 4 VA: x = - 4 (non-removable) hole: (4, 1/5) (remorable) 5) f(x)=\ [[(x ^ 2 - 9)/(x - 3), ifx = 3], [8, ifx = 3]] (x ^ 2 - 9)/(x - 3) = ((x + 3)(x + 3))/(x - 3) = (x + 3) hole (3,6) (removable discontinuity) *not continuons at x = 3 7) f(x)=\ matrix x^ 8 &if x<-1\\ x&& if - 1 <= x < 1 \\ 1-x&&if x>=1 matrix f(- 1) = - 1 lim fa)=1 lim x -1 ,f(a)=-1 lim x -1 +f(x)=-1 x = - 1 lim x 0-1 = f' * (- 1) f( 0 = 0 lim x -> 1 f(x) = sin e Im lim X 0 f ( mathbb O ^ * mathbb Q .: not continww x = 1 (duontinuity) 6) f(x)=\ matrix 2x-3&if x<=2\\ x^ 2 & ifx >2. matrix f(z) = z(x) - 3 f(2) = 1 lim x -> 2 f(x) = dne lim x overline T ^ - f(x) = 1 lim x -> 2 (f(x)) / 4 Since lim x -> 2 f(x) does not exist, f(z) is not continuos at x = 2 8) f(x) = x/(|x| - 3) (discontinuity) * not continuous at x = 3 & x = - 3 ffx = 3 ,x==3 (removable) Find the value of "a" and/or "b" for which the function is continuous. 9) F(x)=\ matrix 7x-2\\ a * x ^ 2 matrix x <= 1 x > 1 7x - 2 = a * x ^ 2 7(1) - z = 0 * (1) ^ 2 50 10) f(x)= alpha x^ 2 &if x<=2\\ 2x+a& ifx >2. ax ^ 2 = Zx + 0x alpha = 4/3 x * (2) ^ 2 = 2(2) + 0 = 4a = 4 + 0 3a = 4 ↑ 11) f(x)=\ [[x + 1, ifx < 1], [ex + b, if * 1 <= x < 2], [3x, ifx >= 2]] x + i = ax + b (1) + 1 = 0(1) + b z = a + p ax + b = 3x a(2) + b = 3(2) 2a + b = 6 a = 2 - b 2(2 - b) + b = 6 4-26+6=6 4 - b = 6 - p - 2 b = - 2 alpha = 2 - (- 2) a=4 Are the following functions continuous at all points in the natural domain? If the function is not continuous, what kind of discontinuity exists? Rewrite the function to make it continuous. 12) f(x) = (x ^ 2 - 16)/(x + 4) = (x + sqrt(4) * (x - 4))/(x + 14) = x - 4 hale (point discontinuity) g(x)= x^ x -146 x+4 &x ne-4\\ -8& x = - 4 13) f(x) = (2x ^ 2 - x - 1)/(x - 1) = (2x+1)xx^ prime -1) (x+1) =2x+1 hote (1, 3) (puint ducntinuity) h(x) = 2x^ 2 -x^ -1 x-1 &x ne1\\ 3&x=1. 4) f(x) = (9x ^ 2 - 4)/(3x + 2) = (3x^ 4 +2)(3x-2) (3x42) =3x-2 hole (- 2/3, - 4) (point (point discontiviurity) k(x)= 9x^ 2 -4 3x+2 &x ne^ - 2/3 \\ -4&x= -2 3 15) g(t) = (sin t)/t not continuous at t = 0 lim t -> 0 (sin t)/t = 1 j(t)=\ [[(sin t)/t], [i]] to