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Continuity

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. · 87 questions

8.1 Continuity of a Function

14 q

Solved Examples

Worked · 14
  1. 8.1 SolvedEx.1
    Discuss the continuity of the function f(x)=∣x−3∣f(x) = |x - 3| at x=3x = 3.
  2. 8.1 SolvedEx.2
    Determine whether the function ff is continuous on the set of real numbers where f(x)=3x+1f(x) = 3x + 1, for x<2x < 2 =7= 7, for 2≤x<42 \le x < 4 =x2−8= x^2 - 8 for x≥4x \ge 4. If it is discontinuous, state the type of discontinuity.
  3. 8.1 SolvedEx.3
    Test whether the function f(x)f(x) is continuous at x=−4x = -4, where f(x)=x2+16x+48x+4f(x) = \frac{x^2 + 16x + 48}{x + 4}, for x≠−4x \ne -4 =8= 8, for x=−4x = -4.
  4. 8.1 SolvedEx.4
    Discuss the continuity of f(x)=9−a2f(x) = \sqrt{9 - a^2}, on the interval [−3,3][-3, 3].
  5. 8.1 SolvedEx.5
    Show that the function f(x)=⌊x⌋f(x) = \lfloor x \rfloor is not continuous at x=0,1x = 0, 1 in the interval [−1,2)[-1, 2).
  6. 8.1 SolvedEx.6
    Discuss the continuity of the following function at x=0x = 0, where f(x)=x2sin⁡(1x)f(x) = x^2 \sin\left(\frac{1}{x}\right), for x≠0x \ne 0 =0= 0, for x=0x = 0.
  7. 8.1 SolvedEx.7
    Find kk if f(x)f(x) is continuous at x=0x = 0, where f(x)=xex+tan⁡xsin⁡3xf(x) = \frac{x e^x + \tan x}{\sin 3x}, for x≠0x \ne 0 =k= k, for x=0x = 0.
  8. 8.1 SolvedEx.8
    If ff is continuous at x=1x = 1, where f(x)=sin⁡(πx)x−1+af(x) = \frac{\sin(\pi x)}{x - 1} + a, for x<1x < 1 =2π= 2\pi, for x=1x = 1 =1+cos⁡(πx)π(1−x)2+b= \frac{1 + \cos(\pi x)}{\pi(1 - x)^2} + b, for x>1x > 1, then find the values of aa and bb.
  9. Identify discontinuities for the following functions as either a jump or a removable discontinuity on R.
    8.1 SolvedEx.9(1)
    f(x)=x2−3x−18x−6f(x) = \frac{x^2 - 3x - 18}{x - 6}
  10. 8.1 SolvedEx.9(2)
    g(x)=3x+1g(x) = 3x + 1, for x<3x < 3 =2−3x= 2 - 3x, for x≥3x \ge 3
  11. 8.1 SolvedEx.9(3)
    h(x)=13−x2h(x) = 13 - x^2, for x<5x < 5 =13−5x= 13 - 5x, for x>5x > 5
  12. 8.1 SolvedEx.10
    Show that the function f(x)=5cos⁡x−e(π2−x)cot⁡xf(x) = \dfrac{5^{\cos x} - e^{\left(\frac{\pi}{2} - x\right)}}{\cot x}, for x≠π2x \ne \dfrac{\pi}{2} =log⁡5−e= \log 5 - e, for x=π2x = \dfrac{\pi}{2} has a removable discontinuity at x=π2x = \dfrac{\pi}{2}. Redefine the function so that it becomes continuous at x=π2x = \dfrac{\pi}{2}.
  13. 8.1 SolvedEx.11
    If f(x)=(3x+22−5x)1xf(x) = \left( \dfrac{3x + 2}{2 - 5x} \right)^{\frac{1}{x}}, for x≠0x \ne 0, is continuous at x=0x = 0 then find f(0)f(0).
  14. 8.1 SolvedEx.12
    If f(x)f(x) is defined on RR, discuss the continuity of ff at x=π2x = \dfrac{\pi}{2}, where f(x)=5cos⁡x+5−cos⁡x−2(3cot⁡x).log⁡(2+π−2x2)f(x) = \dfrac{5^{\cos x} + 5^{-\cos x} - 2}{(3\cot x).\log\left(\dfrac{2 + \pi - 2x}{2}\right)}, for x≠π2x \ne \dfrac{\pi}{2} =2log⁡53= \dfrac{2\log 5}{3}, for x=π2x = \dfrac{\pi}{2}.

Exercise 8.1

43 q
  1. Examine the continuity of
    Ex 8.1 Q1(i)
    f(x)=x3+2x2−x−2f(x) = x^{3} + 2x^{2} - x - 2 at x=−2x = -2.
  2. Ex 8.1 Q1(ii)
    f(x)=sin⁡xf(x) = \sin x, for x≤π4x \le \dfrac{\pi}{4} =cos⁡x= \cos x, for x>π4x > \dfrac{\pi}{4}, at x=π4x = \dfrac{\pi}{4}
  3. Ex 8.1 Q1(iii)
    f(x)=x2−9x−3f(x) = \dfrac{x^{2} - 9}{x - 3}, for x≠3x \ne 3 =8= 8 for x=3x = 3
  4. Examine whether the function is continuous at the points indicated against them.
    Ex 8.1 Q2(i)
    f(x)=x3−2x+1f(x) = x^{3} - 2x + 1, if x≤2x \le 2 =3x−2= 3x - 2, if x>2x > 2, at x=2x = 2.
  5. Ex 8.1 Q2(ii)
    f(x)=x2+18x−19x−1f(x) = \dfrac{x^{2} + 18x - 19}{x - 1}, for x≠1x \ne 1 =20= 20 for x=1x = 1, at x=1x = 1
  6. Ex 8.1 Q2(iii)
    f(x)=xtan⁡3x+2f(x) = \dfrac{x}{\tan 3x} + 2, for x<0x < 0 =73= \dfrac{7}{3}, for x≥0x \ge 0, at x=0x = 0.
  7. Ex 8.1 Q3
    Find all the points of discontinuities of f(x)=⌊x⌋f(x) = \lfloor x \rfloor on the interval (−3,2)(-3, 2).
  8. Ex 8.1 Q4
    Discuss the continuity of the function f(x)=∣2x+3∣f(x) = |2x + 3|, at x=−3/2x = -3/2
  9. Test the continuity of the following functions at the points or interval indicated against them.
    Ex 8.1 Q5(i)
    f(x)=x−1−(x−1)13x−2f(x) = \dfrac{\sqrt{x - 1} - (x - 1)^{\frac{1}{3}}}{x - 2}, for x≠2x \ne 2 =15= \dfrac{1}{5}, for x=2x = 2 at x=2x = 2
  10. Ex 8.1 Q5(ii)
    f(x)=x3−8x+2−3x−2f(x) = \dfrac{x^{3} - 8}{\sqrt{x + 2} - \sqrt{3x - 2}} for x≠2x \ne 2 =−24= -24 for x=2x = 2, at x=2x = 2
  11. Ex 8.1 Q5(iii)
    f(x)=4x+1f(x) = 4x + 1, for x≤83x \le \dfrac{8}{3} =59−9x3= \dfrac{59 - 9x}{3}, for x>83x > \dfrac{8}{3}, at x=83x = \dfrac{8}{3}.
  12. Ex 8.1 Q5(iv)
    f(x)=(27−2x)13−39−3(243+5x)15f(x) = \dfrac{(27 - 2x)^{\frac{1}{3}} - 3}{9 - 3(243 + 5x)^{\frac{1}{5}}}, for x≠0x \ne 0 =2= 2 for x=0x = 0, at x=0x = 0
  13. Ex 8.1 Q5(v)
    f(x)=x2+8x−202x2−9x+10f(x) = \dfrac{x^{2} + 8x - 20}{2x^{2} - 9x + 10} for 0<x<30 < x < 3; x≠2x \ne 2 =12= 12, for x=2x = 2 =2−2x−x2x−4= \dfrac{2 - 2x - x^{2}}{x - 4} for 3≤x<43 \le x < 4 at x=2x = 2
  14. Identify discontinuities for the following functions as either a jump or a removable discontinuity.
    Ex 8.1 Q6(i)
    f(x)=x2−10x+21x−7f(x) = \dfrac{x^{2} - 10x + 21}{x - 7}.
  15. Ex 8.1 Q6(ii)
    f(x)=x2+3x−2f(x) = x^{2} + 3x - 2, for x≤4x \le 4 =5x+3= 5x + 3, for x>4x > 4.
  16. Ex 8.1 Q6(iii)
    f(x)=x2−3x−2f(x) = x^{2} - 3x - 2, for x<−3x < -3 =3+8x= 3 + 8x, for x>−3x > -3.
  17. Ex 8.1 Q6(iv)
    f(x)=4+sin⁡xf(x) = 4 + \sin x, for x<πx < \pi =3−cos⁡x= 3 - \cos x for x>πx > \pi
  18. Show that following functions have continuous extension to the point where f(x)f(x) is not defined. Also find the extension.
    Ex 8.1 Q7(i)
    f(x)=1−cos⁡2xsin⁡xf(x) = \dfrac{1 - \cos 2x}{\sin x}, for x≠0x \ne 0.
  19. Ex 8.1 Q7(ii)
    f(x)=3sin⁡2x+2cos⁡x(1−cos⁡2x)2(1−cos⁡2x)f(x) = \dfrac{3\sin^{2} x + 2\cos x(1 - \cos 2x)}{2\left(1 - \cos^{2} x\right)}, for x≠0x \ne 0.
  20. Ex 8.1 Q7(iii)
    f(x)=x2−1x3+1f(x) = \dfrac{x^{2} - 1}{x^{3} + 1} for x≠−1x \ne -1.
  21. Discuss the continuity of the following functions at the points indicated against them.
    Ex 8.1 Q8(i)
    f(x)=3−tan⁡xπ−3xf(x) = \dfrac{\sqrt{3} - \tan x}{\pi - 3x}, x≠π3x \ne \dfrac{\pi}{3} =34= \dfrac{3}{4}, for x=π3x = \dfrac{\pi}{3}, at x=π3x = \dfrac{\pi}{3}.
  22. Ex 8.1 Q8(ii)
    f(x)=e1/x−1e1/x+1f(x) = \dfrac{e^{1/x} - 1}{e^{1/x} + 1}, for x≠0x \ne 0 =1= 1, for x=0x = 0, at x=0x = 0.
  23. Ex 8.1 Q8(iii)
    f(x)=4x−2x+1+11−cos⁡2xf(x) = \dfrac{4^{x} - 2^{x + 1} + 1}{1 - \cos 2x}, for x≠0x \ne 0 =(log⁡2)22= \dfrac{(\log 2)^{2}}{2}, for x=0x = 0, at x=0x = 0.
  24. Which of the following functions has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it becomes continuous.
    Ex 8.1 Q9(i)
    f(x)=e5sin⁡x−e2x5tan⁡x−3xf(x) = \dfrac{e^{5\sin x} - e^{2x}}{5\tan x - 3x}, for x≠0x \ne 0 =3/4= 3/4, for x=0x = 0, at x=0x = 0.
  25. Ex 8.1 Q9(ii)
    f(x)=log⁡(1+3x)(1+5x)f(x) = \log_{(1 + 3x)}(1 + 5x) for x>0x > 0 =32x−18x−1= \dfrac{32^{x} - 1}{8^{x} - 1}, for x<0x < 0, at x=0x = 0.
  26. Ex 8.1 Q9(iii)
    f(x)=(3−8x3−2x)1xf(x) = \left( \dfrac{3 - 8x}{3 - 2x} \right)^{\frac{1}{x}}, for x≠0x \ne 0.
  27. Ex 8.1 Q9(iv)
    f(x)=3x+2f(x) = 3x + 2, for −4≤x≤−2-4 \le x \le -2 =2x−3= 2x - 3, for −2<x≤6-2 < x \le 6.
  28. Ex 8.1 Q9(v)
    f(x)=x3−8x2−4f(x) = \dfrac{x^{3} - 8}{x^{2} - 4}, for x>2x > 2 =3= 3, for x=2x = 2 =e3(x−2)2−12(x−2)2= \dfrac{e^{3(x - 2)^{2}} - 1}{2(x - 2)^{2}}, for x<2x < 2
  29. Ex 8.1 Q10(i)
    If f(x)=2+sin⁡x−3cos⁡2xf(x) = \dfrac{\sqrt{2 + \sin x} - \sqrt{3}}{\cos^{2} x}, for x≠π2x \ne \dfrac{\pi}{2}, is continuous at x=π2x = \dfrac{\pi}{2} then find f(π2)f\left(\dfrac{\pi}{2}\right).
  30. Ex 8.1 Q10(ii)
    If f(x)=cos⁡2x−sin⁡2x−13x2+1−1f(x) = \dfrac{\cos^{2} x - \sin^{2} x - 1}{\sqrt{3x^{2} + 1} - 1} for x≠0x \ne 0, is continuous at x=0x = 0 then find f(0)f(0).
  31. Ex 8.1 Q10(iii)
    If f(x)=4x−π+4π−x−2(x−π)2f(x) = \dfrac{4^{x - \pi} + 4^{\pi - x} - 2}{(x - \pi)^{2}} for x≠πx \ne \pi, is continuous at x=πx = \pi, then find f(π)f(\pi).
  32. Ex 8.1 Q11(i)
    If f(x)=24x−8x−3x+112x−4x−3x+1f(x) = \dfrac{24^{x} - 8^{x} - 3^{x} + 1}{12^{x} - 4^{x} - 3^{x} + 1}, for x≠0x \ne 0 =k= k, for x=0x = 0 is continuous at x=0x = 0, find kk.
  33. Ex 8.1 Q11(ii)
    If f(x)=5x+5−x−2x2f(x) = \dfrac{5^{x} + 5^{-x} - 2}{x^{2}}, for x≠0x \ne 0 =k= k for x=0x = 0 is continuous at x=0x = 0, find kk.
  34. Ex 8.1 Q11(iii)
    If f(x)=sin⁡2x5x−af(x) = \dfrac{\sin 2x}{5x} - a, for x>0x > 0 =4= 4 for x=0x = 0 =x2+b−3= x^{2} + b - 3, for x<0x < 0 is continuous at x=0x = 0, find aa and bb.
  35. Ex 8.1 Q11(iv)
    For what values of aa and bb is the function f(x)=ax+2b+18f(x) = ax + 2b + 18, for x≤0x \le 0 =x2+3a−b= x^{2} + 3a - b, for 0<x≤20 < x \le 2 =8x−2= 8x - 2, for x>2x > 2, continuous for every xx?
  36. Ex 8.1 Q11(v)
    For what values of aa and bb is the function f(x)=x2−4x−2f(x) = \dfrac{x^{2} - 4}{x - 2}, for x<2x < 2 =ax2−bx+3= ax^{2} - bx + 3, for 2≤x<32 \le x < 3 =2x−a+b= 2x - a + b, for x≥3x \ge 3 continuous for every xx on RR?
  37. Ex 8.1 Q12
    Discuss the continuity of ff on its domain, where f(x)=∣x+1∣f(x) = |x + 1|, for −3≤x≤2-3 \le x \le 2 =∣x−5∣= |x - 5|, for 2<x≤72 < x \le 7.
  38. Ex 8.1 Q13
    Discuss the continuity of f(x)f(x) at x=π4x = \dfrac{\pi}{4} where, f(x)=(sin⁡x+cos⁡x)3−22sin⁡2x−1f(x) = \dfrac{(\sin x + \cos x)^{3} - 2\sqrt{2}}{\sin 2x - 1}, for x≠π4x \ne \dfrac{\pi}{4} =32= \dfrac{3}{\sqrt{2}}, for x=π4x = \dfrac{\pi}{4}.
  39. Ex 8.1 Q14
    Determine the values of pp and qq such that the following function is continuous on the entire real number line. f(x)=x+1f(x) = x + 1, for 1<x<31 < x < 3 =x2+px+q= x^{2} + px + q, for ∣x−2∣≥1|x - 2| \ge 1.
  40. Ex 8.1 Q15
    Show that there is a root for the equation 2x3−x−16=02x^{3} - x - 16 = 0 between 2 and 3.
  41. Ex 8.1 Q16
    Show that there is a root for the equation x3−3x=0x^{3} - 3x = 0 between 1 and 2.
  42. Ex 8.1 Q17
    Activity : Let f(x)=ax+bf(x) = ax + b (where aa and bb are unknown) =x2+5= x^{2} + 5 for x∈Rx \in R Find the values of aa and bb, so that f(x)f(x) is continuous at x=1x = 1. (Fig. 8.11)
  43. Ex 8.1 Q18
    Activity : Suppose f(x)=px+3f(x) = px + 3 for a≤x≤ba \le x \le b =5x2−q= 5x^{2} - q for b<x≤cb < x \le c Find the condition on pp, qq, so that f(x)f(x) is continuous on [a,c][a, c], by filling in the boxes. f(b)=□f(b) = \square lim⁡x→b+f(x)=□\lim_{x \to b^{+}} f(x) = \square ∴pb+3=□−q\therefore pb + 3 = \square - q ∴p=□b\therefore p = \dfrac{\square}{b} is the required condition.

Miscellaneous Exercise 8

30 q

(I) Select the correct answer

Practice · 10
  1. Misc I Q1
    f(x)=2cot⁡x−1π−2xf(x)=\frac{2^{\cot x}-1}{\pi-2x}, for x≠π2x \ne \frac{\pi}{2} =log⁡2=\log\sqrt{2}, for x=π2x=\frac{\pi}{2}

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  2. Misc I Q2
    If f(x)=1−2sin⁡xπ−4xf(x)=\frac{1-\sqrt{2}\sin x}{\pi-4x}, for x≠π4x \ne \frac{\pi}{4} is continuous at x=π4x=\frac{\pi}{4}, then f(π4)=f\left(\frac{\pi}{4}\right)=

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  3. Misc I Q3
    If f(x)=(sin⁡2x)tan⁡5x(e2x−1)2f(x)=\frac{(\sin 2x)\tan 5x}{(e^{2x}-1)^2}, for x≠0x \ne 0 is continuous at x=0x=0, then f(0)f(0) is

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  4. Misc I Q4
    f(x)=x2−7x+10x2+2x−8f(x)=\frac{x^2-7x+10}{x^2+2x-8}, for x∈[−6,−3]x \in [-6,-3]

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  5. Misc I Q5
    If f(x)=ax2+bx+1f(x)=ax^2+bx+1, for ∣x−1∣≥3|x-1| \ge 3 and =4x+5=4x+5, for −2<x<4-2 < x < 4 is continuous everywhere then,

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  6. Misc I Q6
    f(x)=(16x−1)(9x−1)(27x−1)(32x−1)f(x)=\frac{(16^x-1)(9^x-1)}{(27^x-1)(32^x-1)}, for x≠0x \ne 0 =k=k, for x=0x=0 is continuous at x=0x=0, then 'kk' ==

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  7. Misc I Q7
    f(x)=32x−8x−4x+14x−2x+1+1f(x)=\frac{32^x-8^x-4^x+1}{4^x-2^{x+1}+1}, for x≠0x \ne 0 =k=k, for x=0x=0, is continuous at x=0x=0, then value of 'kk' is

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  8. Misc I Q8
    If f(x)=12x−4x−3x+11−cos⁡2xf(x)=\frac{12^x-4^x-3^x+1}{1-\cos 2x}, for x≠0x \ne 0 is continuous at x=0x=0 then the value of f(0)f(0) is

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  9. Misc I Q9
    If f(x)=(4+5x4−7x)4xf(x)=\left(\frac{4+5x}{4-7x}\right)^{\frac{4}{x}}, for x≠0x \ne 0 and f(0)=kf(0)=k, is continuous at x=0x=0, then kk is

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  10. Misc I Q10
    If f(x)=⌊x⌋f(x)=\lfloor x \rfloor for x∈(−1,2)x \in (-1,2) then ff is discontinuous at

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(II) Discuss the continuity

Practice · 7
  1. Misc II Q1
    Discuss the continuity of the following function at the point(s) or on the interval indicated against it. f(x)=x2−3x−10x−5f(x)=\frac{x^2-3x-10}{x-5}, for 3≤x≤6, x≠53 \le x \le 6,\ x \ne 5 =10=10, for x=5x=5 =x2−3x−10x−5=\frac{x^2-3x-10}{x-5}, for 6<x≤96 < x \le 9
  2. Misc II Q2
    Discuss the continuity of the following function at the point(s) or on the interval indicated against it. f(x)=2x2−2x+5f(x)=2x^2-2x+5, for 0≤x≤20 \le x \le 2 =1−3x−x21−x=\frac{1-3x-x^2}{1-x}, for 2<x<42 < x < 4 =x2−25x−5=\frac{x^2-25}{x-5}, for 4≤x≤74 \le x \le 7 and x≠5x \ne 5 =7=7 for x=5x=5
  3. Misc II Q3
    Discuss the continuity of the following function at the point(s) or on the interval indicated against it. f(x)=cos⁡4x−cos⁡9x1−cos⁡xf(x)=\frac{\cos 4x-\cos 9x}{1-\cos x}, for x≠0x \ne 0 f(0)=6815f(0)=\frac{68}{15}, at x=0x=0 on −π2≤x≤π2-\frac{\pi}{2} \le x \le \frac{\pi}{2}
  4. Misc II Q4
    Discuss the continuity of the following function at the point(s) or on the interval indicated against it. f(x)=sin⁡2πx3(1−x)2f(x)=\frac{\sin^2 \pi x}{3(1-x)^2}, for x≠1x \ne 1 =π2sin⁡2(πx2)3+4cos⁡2(πx2)=\frac{\pi^2 \sin^2\left(\frac{\pi x}{2}\right)}{3+4\cos^2\left(\frac{\pi x}{2}\right)} for x=1x=1, at x=1x=1.
  5. Misc II Q5
    Discuss the continuity of the following function at the point(s) or on the interval indicated against it. f(x)=∣x+1∣2x2+x−1f(x)=\frac{|x+1|}{2x^2+x-1}, for x≠−1x \ne -1 =0=0 for x=−1x=-1 at x=−1x=-1.
  6. Misc II Q6
    Discuss the continuity of the following function at the point(s) or on the interval indicated against it. f(x)=[x+1]f(x)=[x+1] for x∈[−2,2)x \in [-2,2) Where [∗][*] is greatest integer function.
  7. Misc II Q7
    Discuss the continuity of the following function at the point(s) or on the interval indicated against it. f(x)=2x2+x+1f(x)=2x^2+x+1, for ∣x−3∣≥2|x-3| \ge 2 =x2+3=x^2+3, for 1<x<51 < x < 5

(III) Identify discontinuities

Practice · 3
  1. Misc III Q1
    Identify discontinuities if any for the following function as either a jump or a removable discontinuity on its respective domain. f(x)=x2+x−3f(x)=x^2+x-3, for x∈[−5,−2)x \in [-5,-2) =x2−5=x^2-5, for x∈(−2,5]x \in (-2,5]
  2. Misc III Q2
    Identify discontinuities if any for the following function as either a jump or a removable discontinuity on its respective domain. f(x)=x2+5x+1f(x)=x^2+5x+1, for 0≤x≤30 \le x \le 3 =x3+x+5=x^3+x+5, for 3<x≤63 < x \le 6
  3. Misc III Q3
    Identify discontinuities if any for the following function as either a jump or a removable discontinuity on its respective domain. f(x)=x2+x+1x+1f(x)=\frac{x^2+x+1}{x+1}, for x∈[0,3)x \in [0,3) =3x+4x2−5=\frac{3x+4}{x^2-5}, for x∈[3,6]x \in [3,6].

(IV) Identify, classify and redefine

Practice · 2
  1. Misc IV Q1
    Discuss the continuity of the following function at the point or on the interval indicated against it. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous. f(x)=(x+3)(x2−6x+8)x2−x−12f(x)=\frac{(x+3)(x^2-6x+8)}{x^2-x-12}
  2. Misc IV Q2
    Discuss the continuity of the following function at the point or on the interval indicated against it. If the function is discontinuous, identify the type of discontinuity and state whether the discontinuity is removable. If it has a removable discontinuity, redefine the function so that it becomes continuous. f(x)=x2+2x+5f(x)=x^2+2x+5, for x≤3x \le 3 =x3−2x2−5=x^3-2x^2-5, for x>3x > 3

(V) Find k

Practice · 2
  1. Misc V Q1
    Find kk if the following function is continuous at the point indicated against it. f(x)=(5x−88−3x)32x−4f(x)=\left(\frac{5x-8}{8-3x}\right)^{\frac{3}{2x-4}}, for x≠2x \ne 2 =k=k, for x=2x=2 at x=2x=2.
  2. Misc V Q2
    Find kk if the following function is continuous at the point indicated against it. f(x)=45x−9x−5x+1(kx−1)(3x−1)f(x)=\frac{45^x-9^x-5^x+1}{(k^x-1)(3^x-1)}, for x≠0x \ne 0 =23=\frac{2}{3}, for x=0x=0, at x=0x=0

(VI) Find a and b

Practice · 2
  1. Misc VI Q1
    Find aa and bb if the following function is continuous at the point or on the interval indicated against it. f(x)=4tan⁡x+5sin⁡xax−1f(x)=\frac{4\tan x+5\sin x}{a^x-1}, for x<0x < 0 =9log⁡2=\frac{9}{\log 2}, for x=0x=0 =11x+7x⋅cos⁡xbx−1=\frac{11x+7x\cdot\cos x}{b^x-1}, for x>0x > 0.
  2. Misc VI Q2
    Find aa and bb if the following function is continuous at the point or on the interval indicated against it. f(x)=ax2+bx+1f(x)=ax^2+bx+1, for ∣2x−3∣≥2|2x-3| \ge 2 =3x+2=3x+2, for 12<x<52\frac{1}{2} < x < \frac{5}{2}.

(VII) Find f(a)

Practice · 2
  1. Misc VII Q1
    Find f(a)f(a), if ff is continuous at x=ax=a where, f(x)=1+cos⁡(πx)π(1−x)2f(x)=\frac{1+\cos(\pi x)}{\pi(1-x)^2}, for x≠1x \ne 1 and at a=1a=1.
  2. Misc VII Q2
    Find f(a)f(a), if ff is continuous at x=ax=a where, f(x)=1−cos⁡[7(x−π)]5(x−π)2f(x)=\frac{1-\cos[7(x-\pi)]}{5(x-\pi)^2}, for x≠πx \ne \pi at a=πa=\pi.

(VIII) Intermediate Value Theorem

Practice · 2
  1. Misc VIII Q1
    Solve using intermediate value theorem. Show that 5x−6x=05^x-6x=0 has a root in [1,2][1,2].
  2. Misc VIII Q2
    Solve using intermediate value theorem. Show that x3−5x2+3x+6=0x^3-5x^2+3x+6=0 has at least two real roots between x=1x=1 and x=5x=5.