Limits and Continuity: Part 10: Tutorial Problems for IITJEE Maths

Problem 1:

The point of discontinuity of the function:

f(x) = \frac{1}{\sin{x} - \cos{x}} in the closed interval [0, \frac{\pi}{2}] are:

(a) 0 and \frac{\pi}{2} (b) \frac{\pi}{2} and \frac{\pi}{4}

(c) \frac{\pi}{4} and 0 (d) \frac{\pi}{4}

Problem 2:

Given f(x) = \frac{x^{2}-9}{x-3} for 0 \leq x <3 and f(x) = 4x-5 for 3 \leq x \leq 6

Consider:

(i) f(x) is discontinuous in (0,3)

(ii) f(x) is discontinuous in (3,6)

(iii) f(x) is continuous in [0,6]

(iv) \lim_{x \rightarrow 3} f(x) exists.

Which of the above statements are false?

(a) only 1 and 2 (b) only 2 and 4 (c) only 1 and 3 (d) none of a, b, c

Problem 3:

For the following function:

f(x) = \frac{x^{2}-3x+2}{x-3} for 0 \leq x \leq 4, and

f(x) = \frac{x^{2}+1}{x-2} for 4 < x \leq 6

Consider

(i) f(x) is discontinuous in (0,4)

(ii) f(x) is discontinuous in (4,6)

(iii) f(x) is discontinuous in [0,6]

(iv) \lim_{x \rightarrow 3}f(x) exists

Which of the above statements are true?

(a) only 1 and 2 (b) only 2 and 4 (c) only 1 and 3 (d) none of a, b and c

Problem 4:

If the function f(x) where

f(x) = \frac{(3^{x}-1)^{2}}{\tan{x} \log{(1+x)}} for x \neq 0

f(x) = \log{k} . \log{\sqrt{3}} for x=0

is continuous at x=0, then k=

(a) 6 (b) \sqrt{3} (c) 9 (d) \frac{3}{2}

Problem 5:

At x = \frac{3 \pi}{4}, the function f(x) where

\frac{\cos{x} + \sin{x}}{3\pi -4x} , where x \neq \frac{3\pi}{4}

f(x) = \frac{1}{\sqrt{2}} where x = \frac{3\pi}{4}

has

(a) removable discontiuity

(b) irremovable discontinuity

(c) no discontinuity

(d) none

Problem 6:

If f(x) is given to be continuous at x=0, where

f(x) = \frac{(e^{kx}-1) \sin{(kx)}}{x^{2}} for x \neq 0 and f(0)=4, then the value of k is:

(a) 2 (b) -2 (c) \pm {2} (d) \pm {\sqrt{2}}

Problem 7:

If given function f(x) is continuous at zero and if

f(x) = \frac{4^{x}-2^{x+1}+1}{1-\cos{x}} when x \neq 0 and f(0)=k, then the value of k is :

(a) \frac{1}{2}(\log{2})^{2} (b) 2(\log{2})^{2} (c) 4 \log{2} (d) \frac{1}{4} \log{2}

Problem 8:

If f(x) is continuous at x=3, where

f(x) = \frac{(2^{x}-8) \log{(x-2)}}{1- \cos{(x-3)}} when x \neq 3 and f(3)=k then the value of k is:

(a) 16 \log{2} (b) 4 \log{2} (c) 8\log{2} (d) 2 \log{2}

Problem 9:

A function f(x) is defined as follows:

f(x) = \frac{ab^{x}-ba^{x}}{x^{2}-1} where x \neq 1 and f(1)=k is continuous at x=1, then find the value of k.

Problem 10:

At the point x=0 the function f(x) where

f(x) = \frac{\log{\sec^{2}{(x)}}}{x \sin{x}}, when x \neq 0

f(x) =e when x=0 possesses

(a) removable discontinuity

(b) irremovable discontinuity

(c) no discontinuity

(d) none

Regards,

Nalin Pithwa

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