Reference no: EM132607953

Question 1. Find the limits of the following functions

(i) lim_x→0 e^x - (1 + x)/x²

(ii) lim_x→0 (1-cos x-x²/2)/x⁴

(iii) lim_x→∞ x²(e^-1/x2 -1)

(iv) lim_x→0 (x-|x|)/x

Question 2. Prove the following

(i) If f (x) and g(x) are continuous at x = a then |f| and max(f, g) are also continuous at x = a.

(ii) If f (x) is continuous at x = a and g(x) is continuous at x = f (a) then (gof )(x) =: g(f (x)) is continuous at x = a.

Question 3. Check if the following function are continuous in R. If not, identify the points and types of discontinuity.

(i) x/(x-1)

(ii) x tan x/(x²+1)

(iii) [x]

(iv) χQ

(v) 1/x^p , p > 0

(vi) (e^x-1)/x , where [.] denotes the greates integer function and χQ is the characteristic function over rational, Q.

Question 4. Check for the uniform continuity of the following functions.

(i) f (x) = x³, 0 ≤ x ≤ 1 (ii) f (x) = x³, 0 ≤ x ≤ ∞ (iii) f (x) = sin x², 0 ≤ x < ∞
(iv) f (x) = 1/1+x² , 0 ≤ x < ∞ (v) e^x2 sin x², 0 ≤ x < ∞ (vi) √x sin x, 0 ≤ x < ∞

Question 5. Check if the following functions are differentiable at x = 0

i) f(x) = xcos1/x, x ≠ 0 f(x)= 0, x = 0

ii) f(x) = g'(x), where g(x) = x² log 1/|x|, x≠ 0 and g(x) = 0, x = 0

Question 6. Prove that f'(0) = 1 for the following function

f (x) = x, x is rational   f (x) = sin x, x is irrational

Question 7. Find the derivative of the function, f (x) = √(x + √(x + √x,)) 0 ≤ x < ∞, if exists.

Question 8. Which of the following functions obey the hypothesis of Rolle's theorem

(i) f (x) = x², 0 ≤ x ≤ 1 (ii) f (x) = sin x, 0 ≤ x ≤ π (iii) √x(x - 1), 0 ≤ x ≤ 1,
and
iv) f(x) = sin 1/x, x ∈ [- 1/Π , 1/Π] - {0}

    f(x) = 0, x = 0

Question 9. Prove the following

(i) If fJ(x) = 0 for all x in an open interval (a, b), then f (x) = C for all x ∈ (a, b), where C is a constant.
(ii) Using Mean Value Theorem, show that | cos x - cos y| ≤ |x - y|.
(iii) If the derivative of f (x) is bounded, then show that f is uniformly continuous.

(a) If f be differentiable on R and |f (x) - f (y)| ≤ |x- y|2 for all x, y ∈ R, then show that f is constant.

Question 10. Classify the critical points of the following functions into local maxima, local minima and saddle points

(i) f (x) = xⁿ, n > 0 (ii) sin x (iii) |x| (iv) 1/|x|²