Limits and Continuity: Part 9: Tutorial Practice for IITJEE Maths

Problem 1:

If f(x) = \frac{(5^{\sin{x}}-1)^{2}}{x \log{(1+2x)}}, where x \neq 0 is continuous at zero, then find the value of f(0).

Problem 2:

If f(x) = 2x + a for 0 \leq x <1 and f(x) = 3x+b for 1 \leq x \leq 2 is continuous at x=1 and a+b=1, then the find the value of 3a-4b.

Problem 3:

If f(x) = \frac{2^{3x}-3^{x}}{x} for x<0 and f(x) = \frac{1}{x} \log{(1+ \frac{8x}{3})} for x>0.

Consider the following statements:

i) \lim_{x \rightarrow 0} f(x) does not exist.

ii) \lim_{x \rightarrow 0^{+}} f(x) exists but f(0) is not defined.

iii) f(x) is discontinuous at zero

iv) \lim_{x \rightarrow 0^{-}} f(x) exists, but f(0) is not defined.

Which of the above statements are false?

(a) all four (b) (ii) and (iv) (c) (i) and (iii) (d) none

Problem 4:

If f(x) = \frac{\log{x} - \log{2}}{x-2} for x >2 and f(x) = \frac{1-\cos{(x-2)}}{(x-2)^{2}} for x <2

Consider the following statements:

(i) \lim_{x \rightarrow 2^{-}} f(x) does not exist.

(ii) \lim_{x \rightarrow 2^{+}} does not exist.

(iii) f(x) is continuous at x=2

(iv) f(x) is discontinuous at x=2.

Which of the above statements are true?

(a) none (b) iv (c) iii (d) ii

Problem 5:

If the function f is continuous at x=0 and is defined by

f(x) = \frac{\sin{4x}}{5x}+a for x>0

f(x) = x+4-b for x <0

f(x) = 1 for x =0

The quadratic equation whose roots are values of 5a and 2b is

(a) x^{2}-2x+3=0 (b) x^{2} + 3x +2=0

(c) x^{2}-3x =2=0 (d) none

Problem 6:

The function f(x) = \frac{(e^{3x}-1)^{2}}{x \log{(1+3x)}} for x \neq 0 and f(0)=\frac{1}{3}

(a) has a removable discontinuity at x=0

(b) has irremovable discontinuity at x=0

(c) is continuous at x=0

(d) none of the above.

Problem 7:

If f(x) is continuous in [0,8] and

f(x) = x^{2} + ax + b when 0 \leq x <2

f(x) = 3x+2 when 2 \leq x \leq 4

f(x) = 2ax + 5b when 4 < x \leq 8

Then, values of a and b are:

(a) 3,2 (b) 1, -2 (c) -3, 2 (d) 3,-2

Problem 8:

The value of a^{2} - b^{2} if f is continuous on [-\pi, \pi] where

f(x) = -2\sin{x} for -\pi \leq x \leq -\frac{\pi}{2}

f(x) = a \sin{x} + b for -\frac{\pi}{2} < x < \frac{\pi}{2}

f(x) = \cos{x} for \frac{\pi}{2} \leq x \leq \pi is

(a) 0 (b) 2 (c) \infty (d) indeterminate

Problem 9:

Given f(x) = \frac{x^{2}+3x+5}{x^{2}-7x+10}. Let A \equiv [-2,3] and B \equiv [6,10] then

(a) f(x) is continuouis in B but discontinuous in A

(b) f(x) is discontinuous in B but continuous in A

(c) f(x) is continuous in both A and B

(d) f(x) is discontinuous in both A and B

Problem 10:

The function f(x) = \frac{2x^{2}-x+7}{x^{2}-4x+5} is

(a) continuous for all real values of x

(b) discontinuous for all real values of x

(c) discontinuous at x=1 and x=5

(d) discontinuous at x=2 and x=3

Regards,

Nalin Pithwa

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