Define functions f,g: RR by: f(x) = {rsin(2), if x 40 and 9(2) 2sin(), if x...

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Define functions f,g: RR by: f(x) = {rsin(2), if x 40 and 9(2) 2sin(), if x 0 = if x=0 if x=0. (a) Prove that f'(0) does not exist. (b) Prove that g(x) is differentiable on R, and g'(x) is not continuous at x=0. Problem 3: Let r2sin(), if x is an irrational number f(x)= = {siu Show that f'(0) exists. if x is a rational number C. Problem 4: Let f.g be functions which are both differentiable at x=c. Moreover, assume that f(c) = g(c)=0 and g'(c) 0. Prove that lim f(x) f'(c) g(z) = g'(c) (Note that this is not L'Hospital's rule.) Define functions f,g: RR by: f(x) = {rsin(2), if x 40 and 9(2) 2sin(), if x 0 = if x=0 if x=0. (a) Prove that f'(0) does not exist. (b) Prove that g(x) is differentiable on R, and g'(x) is not continuous at x=0. Problem 3: Let r2sin(), if x is an irrational number f(x)= = {siu Show that f'(0) exists. if x is a rational number C. Problem 4: Let f.g be functions which are both differentiable at x=c. Moreover, assume that f(c) = g(c)=0 and g'(c) 0. Prove that lim f(x) f'(c) g(z) = g'(c) (Note that this is not L'Hospital's rule.)