(b) Find the value of a for which f(2) is continuous at x = -1, where (+31 + 2 2+1 f(0) = if x + -1, if x = -1. le Answer:

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20 Sep 2018

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(2+21=3 3 5. Let f(c)= -11 if : 71 Find all x where f() is continuous. if r=1 14 Solution: By the continuity theorems we have that f() is continuous for all 1 #1. For r = 1, we have = lim 11- -(x+3) = -4 11- 1 + 2.0 -3 : lim lim (+3)(x - 1) f(1) = 2-1 1-1 1-1- -(2-1) Thus, lim f(1) = f(1), so f(t) is not continuous at x = 1. Therefore, f(t) is continuous on (-0,1) U (1,0).

[8] 9. Either prove the statement is true or show that it is false. (a) sin (sin(2)) = 27. Solution: It is FALSE. The range of sin-' r is (-1,1], hence sin (sin(27)) = sin (0) = 0 (b) If f is continuous at a, then f is continuous at a. Solution: It is TRUE. Let g(x) = \f()]. If lim f(x) = f(a), then we have since the absolute value function is continuous lim g(x) = lim f() = lim f(x) = f(a) = g(a) So, g = \f is continuous at a. (c) If [g] is continuous at a, then g is continuous at a. -1 if r <0 Solution: It is FALSE. Let g(x) = 3 · Then, g = 1, so g is continuous at 0 i if >0 by the Continuity Theorems, but lim g(t) = -1 + lim g(1) 240- 1- 0+ So, g(x) is not continuous at a. (d) If f is continuous at a, then f is differentiable at a. Solution: It is FALSE. We know that f(1) = 2 is continuous at 0, but is not differentiable at 0.

is the graph shown in figure. Find the value of that satisfies the Rolle’s Theorem for on the interval .

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(b) Find the value of a for which f(2) is continuous at x = -1, where (+31 + 2 2+1 f(0) = if x + -1, if x = -1. le Answer:

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