Page 1 WORKSHEET II Course: Mat101E Topic: Limits and Continuity 1. Use ?-d denition to prove that: (a) lim x?-4 v 1-2x = 3 (b) lim x? v 3 1 x 2 = 1 3 (c) lim x?2 x 2 -1 x+3 = 3 5 (d) lim x?1 1 v x+1 = 1 2 (e) lim x?2 (x-2) 3 sin 1 x-2 = 0 (f) lim x?1 (x 2 -1)cos 1 x-1 = 0 (g) lim x?1 (x 2 +x+3) = 5 (h) lim x?1 (x 2 +x+3)?= 1 2. For the following limits, nd the appropriate value of d that corresponds to the given ? value: (a) lim x?-1 (1-2x) = 3, ? = 0.01 (b) lim x?0 1 x-1 =-1, ? = 0.5 (c) lim x?2 v 11-x = 3, ? = 1 3. Find the following limits, if they exist, or explain why they do not exist. (a) lim x?64 v x-8 3 v x-4 (b) lim x?1 1- v x 1-x (c) lim x?27 v x-3 v 3 3 v x-3 (d) lim x?0 3 v x+1-1 4 v x+1-1 (e) lim x?0 x tan3x (f) lim x? sinx x-p (g) lim x?0 sin 2 x x(1-cosx) (h) lim x?8 x+sinx x+cosx (i) lim x?±8 tan -1 x (j) lim x?8 v x+x v x+cosx (k) lim x?8 e x sin(e -x ) (l) lim x?8 x 2 +2 x-5 (m) lim x?8 x-5 x 2 +2 (n) lim h?0 cosh-1 h (o) lim x?8 2x sinx (p) lim x?8 x+sinx 2x+5 (q) lim x?0 tan -1 x x (r) lim x?0 1 2+x - 1 2 x (s) lim x?0 xsin 1 x (t) lim x?0 sin 2 3x 5x 2 (u) lim x?0 v 1+tanx- v 1+sinx x 3 (v) lim x?0 v 1+x 2 - v 1+x x (w) lim x?0 1+sinx-cosx 1-sinx-cosx (x) lim x?8 x( v 9x 2 +1-3x) (y) lim x?3 3-x v 4-x- v x 3 (z) lim x?8 ( v x 2 +1- v x 2 -1) 1 Page 2 WORKSHEET II Course: Mat101E Topic: Limits and Continuity 1. Use ?-d denition to prove that: (a) lim x?-4 v 1-2x = 3 (b) lim x? v 3 1 x 2 = 1 3 (c) lim x?2 x 2 -1 x+3 = 3 5 (d) lim x?1 1 v x+1 = 1 2 (e) lim x?2 (x-2) 3 sin 1 x-2 = 0 (f) lim x?1 (x 2 -1)cos 1 x-1 = 0 (g) lim x?1 (x 2 +x+3) = 5 (h) lim x?1 (x 2 +x+3)?= 1 2. For the following limits, nd the appropriate value of d that corresponds to the given ? value: (a) lim x?-1 (1-2x) = 3, ? = 0.01 (b) lim x?0 1 x-1 =-1, ? = 0.5 (c) lim x?2 v 11-x = 3, ? = 1 3. Find the following limits, if they exist, or explain why they do not exist. (a) lim x?64 v x-8 3 v x-4 (b) lim x?1 1- v x 1-x (c) lim x?27 v x-3 v 3 3 v x-3 (d) lim x?0 3 v x+1-1 4 v x+1-1 (e) lim x?0 x tan3x (f) lim x? sinx x-p (g) lim x?0 sin 2 x x(1-cosx) (h) lim x?8 x+sinx x+cosx (i) lim x?±8 tan -1 x (j) lim x?8 v x+x v x+cosx (k) lim x?8 e x sin(e -x ) (l) lim x?8 x 2 +2 x-5 (m) lim x?8 x-5 x 2 +2 (n) lim h?0 cosh-1 h (o) lim x?8 2x sinx (p) lim x?8 x+sinx 2x+5 (q) lim x?0 tan -1 x x (r) lim x?0 1 2+x - 1 2 x (s) lim x?0 xsin 1 x (t) lim x?0 sin 2 3x 5x 2 (u) lim x?0 v 1+tanx- v 1+sinx x 3 (v) lim x?0 v 1+x 2 - v 1+x x (w) lim x?0 1+sinx-cosx 1-sinx-cosx (x) lim x?8 x( v 9x 2 +1-3x) (y) lim x?3 3-x v 4-x- v x 3 (z) lim x?8 ( v x 2 +1- v x 2 -1) 1 4. Let lim x?1 f(x) =-1. Evaluate lim x?1 sin(1+f(x)) 1-f 2 (x) . 5. Find the right-hand and the left-hand limits of the following functions at the given point(s). (a) y = |x-1| x-1 +x 2 , (x = 1) (b) y = 1 3-4 1 x2 ,(x = 2) (c) y = 1 x 4=3 - 1 (x-2) 1=3 , (x = 0,2) (d) y = 2+x 1+2 1=x , (x = 0) (e) y = v 1-cos2x v 2x , (x = 0) (f) y = tan -1 x x-2 ,(x = 2) (g) y = x x 2 -1 +x 2 , (x =±1) (h) y = ? ? ? 1-x 2 , |x|= 1 1 |x| , |x| > 1 , (x =-1) 6. Suppose that f is an even function of x. Does knowing that lim x?2 f(x) = 7 tell you anything about either lim x?-2 f(x) or lim x?-2 + f(x)? Give reasons for your answer. 7. Find the asymptotes, if any, of the following functions. (a) f(x) = x 3 4-x 2 (b) g(x) = x 3 +2x-1 x 3 +2x 2 -x-2 8. (a) Graph the following function f. (b) Find the points, if any, at which f is discontinuous and classify their types. f(x) = ? ? ? ? ? ? ? ? ? ? ? ? ? x+3, -3= x <-1 -1, x =-1 -x+1, -1 < x= 1 1 x-1 , 1 < x= 2 x, x > 2 9. Discuss the limit, one-sided limit, continuity and one-sided continuity of f and g at each of the points x = 0,±1. (a) f(x) = ? ? ? ? ? ? ? ? ? ? ? 1, x=-1 -x, -1 < x < 0 1, x = 0 -x, 0 < x < 1 1, x > 1 (b) g(x) = ? ? ? ? ? ? ? 0, x=-1 1/x, |x| < 1 0, x = 1 1, x > 1 10. For the following functions, nd the discontinuity points, if any, and classify the types of the discontinuities. (a) f(x) = x-2 x+2 (b) f(x) = x 2 +1 x 2 -4x+3 (c) f(x) = 1 x 2 +1 (d) f(x) = |x| x 2 Page 3 WORKSHEET II Course: Mat101E Topic: Limits and Continuity 1. Use ?-d denition to prove that: (a) lim x?-4 v 1-2x = 3 (b) lim x? v 3 1 x 2 = 1 3 (c) lim x?2 x 2 -1 x+3 = 3 5 (d) lim x?1 1 v x+1 = 1 2 (e) lim x?2 (x-2) 3 sin 1 x-2 = 0 (f) lim x?1 (x 2 -1)cos 1 x-1 = 0 (g) lim x?1 (x 2 +x+3) = 5 (h) lim x?1 (x 2 +x+3)?= 1 2. For the following limits, nd the appropriate value of d that corresponds to the given ? value: (a) lim x?-1 (1-2x) = 3, ? = 0.01 (b) lim x?0 1 x-1 =-1, ? = 0.5 (c) lim x?2 v 11-x = 3, ? = 1 3. Find the following limits, if they exist, or explain why they do not exist. (a) lim x?64 v x-8 3 v x-4 (b) lim x?1 1- v x 1-x (c) lim x?27 v x-3 v 3 3 v x-3 (d) lim x?0 3 v x+1-1 4 v x+1-1 (e) lim x?0 x tan3x (f) lim x? sinx x-p (g) lim x?0 sin 2 x x(1-cosx) (h) lim x?8 x+sinx x+cosx (i) lim x?±8 tan -1 x (j) lim x?8 v x+x v x+cosx (k) lim x?8 e x sin(e -x ) (l) lim x?8 x 2 +2 x-5 (m) lim x?8 x-5 x 2 +2 (n) lim h?0 cosh-1 h (o) lim x?8 2x sinx (p) lim x?8 x+sinx 2x+5 (q) lim x?0 tan -1 x x (r) lim x?0 1 2+x - 1 2 x (s) lim x?0 xsin 1 x (t) lim x?0 sin 2 3x 5x 2 (u) lim x?0 v 1+tanx- v 1+sinx x 3 (v) lim x?0 v 1+x 2 - v 1+x x (w) lim x?0 1+sinx-cosx 1-sinx-cosx (x) lim x?8 x( v 9x 2 +1-3x) (y) lim x?3 3-x v 4-x- v x 3 (z) lim x?8 ( v x 2 +1- v x 2 -1) 1 4. Let lim x?1 f(x) =-1. Evaluate lim x?1 sin(1+f(x)) 1-f 2 (x) . 5. Find the right-hand and the left-hand limits of the following functions at the given point(s). (a) y = |x-1| x-1 +x 2 , (x = 1) (b) y = 1 3-4 1 x2 ,(x = 2) (c) y = 1 x 4=3 - 1 (x-2) 1=3 , (x = 0,2) (d) y = 2+x 1+2 1=x , (x = 0) (e) y = v 1-cos2x v 2x , (x = 0) (f) y = tan -1 x x-2 ,(x = 2) (g) y = x x 2 -1 +x 2 , (x =±1) (h) y = ? ? ? 1-x 2 , |x|= 1 1 |x| , |x| > 1 , (x =-1) 6. Suppose that f is an even function of x. Does knowing that lim x?2 f(x) = 7 tell you anything about either lim x?-2 f(x) or lim x?-2 + f(x)? Give reasons for your answer. 7. Find the asymptotes, if any, of the following functions. (a) f(x) = x 3 4-x 2 (b) g(x) = x 3 +2x-1 x 3 +2x 2 -x-2 8. (a) Graph the following function f. (b) Find the points, if any, at which f is discontinuous and classify their types. f(x) = ? ? ? ? ? ? ? ? ? ? ? ? ? x+3, -3= x <-1 -1, x =-1 -x+1, -1 < x= 1 1 x-1 , 1 < x= 2 x, x > 2 9. Discuss the limit, one-sided limit, continuity and one-sided continuity of f and g at each of the points x = 0,±1. (a) f(x) = ? ? ? ? ? ? ? ? ? ? ? 1, x=-1 -x, -1 < x < 0 1, x = 0 -x, 0 < x < 1 1, x > 1 (b) g(x) = ? ? ? ? ? ? ? 0, x=-1 1/x, |x| < 1 0, x = 1 1, x > 1 10. For the following functions, nd the discontinuity points, if any, and classify the types of the discontinuities. (a) f(x) = x-2 x+2 (b) f(x) = x 2 +1 x 2 -4x+3 (c) f(x) = 1 x 2 +1 (d) f(x) = |x| x 2 (e) f(x) = 1 1-3 3x x (f) f(x) = { 1-cosx x 2 , x?= 0 1, x = 0 (g) f(x) = v xsin 1 x (h) f(x) = ? ? ? ? ? ? ? sin -1 x 2 , 0 < x < 2 p, x = 2 tan -1 1 x-2 , x > 2 11. Evaluate the following limit (Do not use the L'H^ opital's Rule). lim x?1 + {ln[sin(x 2 -1)]-ln(x-1)} 12. Dene f(1) in a way that extends f(x) = x 2 +2x-3 x 2 -1 to be continuous at x = 1. 13. For what value of a, is f(x) = { x 2 -1, x < 3 2ax, x= 3 continuous at every x?R? 14. If x 4 = f(x)= x 2 for all x? [-1,1], and x 2 = f(x)= x 4 for all x? (-8,-1)?(1,8), then at which point(s) c do you automatically know lim x?c f(x)? What is the value of the limit at this point(s)? 15. Show that the equation x 3 -2x+2 = 0 must have a solution between-2 and 0. 16. Show that the following functions have at least one real root. (a) f(x) = 3 v x+x-2 (b) g(x) = cosx+sinx-x 17. Suppose that f is a continuous function on the closed interval [0,1] and that 0= f(x)= 1 for every x? [0,1]. Show that there must exist a number c? [0,1] such that f(c) = c. 18. Suppose that f and g are continuous functions on [a,b], and that f(a) < g(a) and f(b) > g(b). Prove that f(c) = g(c) for some c? [a,b]. 19. If F(x) = (x-a) 2 (x-b) 2 +x where a,b? R. Show that there must exist a number c? (a,b) such that F(c) = a+b 2 . 3Read More