Limit And Continuity - Assignment

Pure Mathematics – Limit and Continuity of Functions

p.1

Limit of and Continuity of Functions Assignment 1.

2.

3.

⎧ x 2 + ax + b when x ≤ 1, ⎪ . Let f ( x ) = ⎨ sin πx when x > 1. ⎪ ⎩ π If f is differentiable at 1, find a and b. ⎧ x 2 + bx + c if x ≥ 0, ⎪ Let f ( x ) = ⎨ sin x + 2 x if x < 0. ⎪ ⎩ x (a) If f is continuous at x = 0, find c. (b) If f’(0) exists, find b.

4 + an 2

Let a1 = 1 and a n +1 =

2

for n ∈ ℵ . Show that 1 ≤ a n < 2 for all n ∈ ℵ .

Hence show that {an} is convergent and find its limit. 4.

Let a be a constant and f : ℜ → ℜ be defined by ⎧ x2 − x + a when x < π , ⎪ f ( x ) = ⎨ 2π ⎪a cos x when x ≥ π . ⎩ It is known that f is continuous everywhere. (a) Prove that a =

π 4

.

(b) Prove that f is differentiable at π. (c) Is f’ continuous at π? Explain your answer. 5.

Let f : ℜ → ℜ be a function satisfying the following conditions: (1)

f ( x + y ) = e x f ( y ) + e y f (x ) for all x, y ∈ ℜ ;

(2) lim h →0

f (h ) = 2005 . h

(a) Find f(0). (b) Find lim f (h ) . Hence prove that f is a continuous function. h →0

(c) (i)

Prove that f is differentiable everywhere and that f ' ( x ) = 2005e x + f ( x ) for all x ∈ ℜ .

(ii) Let n be a positive integer. Using (c)(i), find f(n)(0). (d) By considering the derivative of the function

f (x ) , find f(x). ex