Mathematics · Textbook solutions

Limits and Derivatives

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 126 questions

12.3-12.4 Limits

44 q

Solved Examples

Worked · 12
  1. Find the limits:
    12.1 Eg.1(i)
    lim⁡x→1[x3−x2+1]\lim_{x \to 1}\left[x^3 - x^2 + 1\right]
  2. 12.1 Eg.1(ii)
    lim⁡x→3[x(x+1)]\lim_{x \to 3}\left[x(x+1)\right]
  3. 12.1 Eg.1(iii)
    lim⁡x→−1[1+x+x2+…+x10]\lim_{x \to -1}\left[1 + x + x^2 + \ldots + x^{10}\right]
  4. Find the limits:
    12.1 Eg.2(i)
    lim⁡x→1[x2+1x+100]\lim_{x \to 1}\left[\frac{x^2 + 1}{x + 100}\right]
  5. 12.1 Eg.2(ii)
    lim⁡x→2[x3−4x2+4xx2−4]\lim_{x \to 2}\left[\frac{x^3 - 4x^2 + 4x}{x^2 - 4}\right]
  6. 12.1 Eg.2(iii)
    lim⁡x→2[x2−4x3−4x2+4x]\lim_{x \to 2}\left[\frac{x^2 - 4}{x^3 - 4x^2 + 4x}\right]
  7. 12.1 Eg.2(iv)
    lim⁡x→2[x3−2x2x2−5x+6]\lim_{x \to 2}\left[\frac{x^3 - 2x^2}{x^2 - 5x + 6}\right]
  8. 12.1 Eg.2(v)
    lim⁡x→1[x−2x2−x−1x3−3x2+2x]\lim_{x \to 1}\left[\frac{x-2}{x^2 - x} - \frac{1}{x^3 - 3x^2 + 2x}\right]
  9. Evaluate:
    12.1 Eg.3(i)
    lim⁡x→1x15−1x10−1\lim_{x \to 1}\frac{x^{15} - 1}{x^{10} - 1}
  10. 12.1 Eg.3(ii)
    lim⁡x→01+x−1x\lim_{x \to 0}\frac{\sqrt{1+x} - 1}{x}
  11. Evaluate:
    12.1 Eg.4(i)
    lim⁡x→0sin⁡4xsin⁡2x\lim_{x \to 0}\frac{\sin 4x}{\sin 2x}
  12. 12.1 Eg.4(ii)
    lim⁡x→0tan⁡xx\lim_{x \to 0}\frac{\tan x}{x}

Exercise 12.1

Practice · 32
  1. Ex 12.1 Q1
    Evaluate the following limit: lim⁡x→3x+3\lim_{x \to 3} x + 3.
  2. Ex 12.1 Q2
    Evaluate the following limit: lim⁡x→π(x−227)\lim_{x \to \pi} \left( x - \frac{22}{7} \right).
  3. Ex 12.1 Q3
    Evaluate the following limit: lim⁡r→1πr2\lim_{r \to 1} \pi r^2.
  4. Ex 12.1 Q4
    Evaluate the following limit: lim⁡x→44x+3x−2\lim_{x \to 4} \frac{4x + 3}{x - 2}.
  5. Ex 12.1 Q5
    Evaluate the following limit: lim⁡x→−1x10+x5+1x−1\lim_{x \to -1} \frac{x^{10} + x^5 + 1}{x - 1}.
  6. Ex 12.1 Q6
    Evaluate the following limit: lim⁡x→0(x+1)5−1x\lim_{x \to 0} \frac{(x + 1)^5 - 1}{x}.
  7. Ex 12.1 Q7
    Evaluate the following limit: lim⁡x→23x2−x−10x2−4\lim_{x \to 2} \frac{3x^2 - x - 10}{x^2 - 4}.
  8. Ex 12.1 Q8
    Evaluate the following limit: lim⁡x→3x4−812x2−5x−3\lim_{x \to 3} \frac{x^4 - 81}{2x^2 - 5x - 3}.
  9. Ex 12.1 Q9
    Evaluate the following limit: lim⁡x→0ax+bcx+1\lim_{x \to 0} \frac{ax + b}{cx + 1}.
  10. Ex 12.1 Q10
    Evaluate the following limit: lim⁡z→1z13−1z16−1\lim_{z \to 1} \frac{z^{\frac{1}{3}} - 1}{z^{\frac{1}{6}} - 1}.
  11. Ex 12.1 Q11
    Evaluate the following limit: lim⁡x→1ax2+bx+ccx2+bx+a\lim_{x \to 1} \frac{ax^2 + bx + c}{cx^2 + bx + a}, a+b+c≠0a + b + c \neq 0.
  12. Ex 12.1 Q12
    Evaluate the following limit: lim⁡x→−21x+12x+2\lim_{x \to -2} \dfrac{\frac{1}{x} + \frac{1}{2}}{x + 2}.
  13. Ex 12.1 Q13
    Evaluate the following limit: lim⁡x→0sin⁡axbx\lim_{x \to 0} \frac{\sin ax}{bx}.
  14. Ex 12.1 Q14
    Evaluate the following limit: lim⁡x→0sin⁡axsin⁡bx\lim_{x \to 0} \frac{\sin ax}{\sin bx}, a,b≠0a, b \neq 0.
  15. Ex 12.1 Q15
    Evaluate the following limit: lim⁡x→πsin⁡(π−x)π(π−x)\lim_{x \to \pi} \frac{\sin(\pi - x)}{\pi(\pi - x)}.
  16. Ex 12.1 Q16
    Evaluate the following limit: lim⁡x→0cos⁡xπ−x\lim_{x \to 0} \frac{\cos x}{\pi - x}.
  17. Ex 12.1 Q17
    Evaluate the following limit: lim⁡x→0cos⁡2x−1cos⁡x−1\lim_{x \to 0} \frac{\cos 2x - 1}{\cos x - 1}.
  18. Ex 12.1 Q18
    Evaluate the following limit: lim⁡x→0ax+xcos⁡xbsin⁡x\lim_{x \to 0} \frac{ax + x \cos x}{b \sin x}.
  19. Ex 12.1 Q19
    Evaluate the following limit: lim⁡x→0xsec⁡x\lim_{x \to 0} x \sec x.
  20. Ex 12.1 Q20
    Evaluate the following limit: lim⁡x→0sin⁡ax+bxax+sin⁡bx\lim_{x \to 0} \frac{\sin ax + bx}{ax + \sin bx}, a,b,a+b≠0a, b, a + b \neq 0.
  21. Ex 12.1 Q21
    Evaluate the following limit: lim⁡x→0(cosec⁡x−cot⁡x)\lim_{x \to 0} (\operatorname{cosec} x - \cot x).
  22. Ex 12.1 Q22
    Evaluate the following limit: lim⁡x→π2tan⁡2xx−π2\lim_{x \to \frac{\pi}{2}} \dfrac{\tan 2x}{x - \dfrac{\pi}{2}}.
  23. Ex 12.1 Q23
    Find lim⁡x→0f(x)\lim_{x \to 0} f(x) and lim⁡x→1f(x)\lim_{x \to 1} f(x), where f(x)={2x+3,x≤03(x+1),x>0f(x) = \begin{cases} 2x + 3, & x \le 0 \\ 3(x + 1), & x > 0 \end{cases}
  24. Ex 12.1 Q24
    Find lim⁡x→1f(x)\lim_{x \to 1} f(x), where f(x)={x2−1,x≤1−x2−1,x>1f(x) = \begin{cases} x^2 - 1, & x \le 1 \\ -x^2 - 1, & x > 1 \end{cases}
  25. Ex 12.1 Q25
    Evaluate lim⁡x→0f(x)\lim_{x \to 0} f(x), where f(x)={∣x∣x,x≠00,x=0f(x) = \begin{cases} \dfrac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}
  26. Ex 12.1 Q26
    Find lim⁡x→0f(x)\lim_{x \to 0} f(x), where f(x)={x∣x∣,x≠00,x=0f(x) = \begin{cases} \dfrac{x}{|x|}, & x \neq 0 \\ 0, & x = 0 \end{cases}
  27. Ex 12.1 Q27
    Find lim⁡x→5f(x)\lim_{x \to 5} f(x), where f(x)=∣x∣−5f(x) = |x| - 5.
  28. Ex 12.1 Q28
    Suppose f(x)={a+bx,x<14,x=1b−ax,x>1f(x) = \begin{cases} a + bx, & x < 1 \\ 4, & x = 1 \\ b - ax, & x > 1 \end{cases} and if lim⁡x→1f(x)=f(1)\lim_{x \to 1} f(x) = f(1) what are possible values of aa and bb?
  29. Ex 12.1 Q29
    Let a1,a2,…,ana_1, a_2, \ldots, a_n be fixed real numbers and define a function f(x)=(x−a1)(x−a2)…(x−an)f(x) = (x - a_1)(x - a_2) \ldots (x - a_n). What is lim⁡x→a1f(x)\lim_{x \to a_1} f(x)? For some a≠a1,a2,…,ana \neq a_1, a_2, \ldots, a_n, compute lim⁡x→af(x)\lim_{x \to a} f(x).
  30. Ex 12.1 Q30
    If f(x)={∣x∣+1,x<00,x=0∣x∣−1,x>0f(x) = \begin{cases} |x| + 1, & x < 0 \\ 0, & x = 0 \\ |x| - 1, & x > 0 \end{cases} For what value(s) of aa does lim⁡x→af(x)\lim_{x \to a} f(x) exists?
  31. Ex 12.1 Q31
    If the function f(x)f(x) satisfies lim⁡x→1f(x)−2x2−1=π\lim_{x \to 1} \frac{f(x) - 2}{x^2 - 1} = \pi, evaluate lim⁡x→1f(x)\lim_{x \to 1} f(x).
  32. Ex 12.1 Q32
    If f(x)={mx2+n,x<0nx+m,0≤x≤1nx3+m,x>1f(x) = \begin{cases} mx^2 + n, & x < 0 \\ nx + m, & 0 \le x \le 1 \\ nx^3 + m, & x > 1 \end{cases} For what integers mm and nn does both lim⁡x→0f(x)\lim_{x \to 0} f(x) and lim⁡x→1f(x)\lim_{x \to 1} f(x) exist?

12.5 Derivatives

41 q

Solved Examples

Worked · 14
  1. 12.2 Eg.5
    Find the derivative at x=2x = 2 of the function f(x)=3xf(x) = 3x.
  2. 12.2 Eg.6
    Find the derivative of the function f(x)=2x2+3x−5f(x) = 2x^2 + 3x - 5 at x=−1x = -1. Also prove that f′(0)+3f′(−1)=0f'(0) + 3f'(-1) = 0.
  3. 12.2 Eg.7
    Find the derivative of sin⁡x\sin x at x=0x = 0.
  4. 12.2 Eg.8
    Find the derivative of f(x)=3f(x) = 3 at x=0x = 0 and at x=3x = 3.
  5. 12.2 Eg.9
    Find the derivative of f(x)=10xf(x) = 10x.
  6. 12.2 Eg.10
    Find the derivative of f(x)=x2f(x) = x^2.
  7. 12.2 Eg.11
    Find the derivative of the constant function f(x)=af(x) = a for a fixed real number aa.
  8. 12.2 Eg.12
    Find the derivative of f(x)=1xf(x) = \frac{1}{x}.
  9. 12.2 Eg.13
    Compute the derivative of 6x100−x55+x6x^{100} - x^{55} + x.
  10. 12.2 Eg.14
    Find the derivative of f(x)=1+x+x2+x3+…+x50f(x) = 1 + x + x^2 + x^3 + \ldots + x^{50} at x=1x = 1.
  11. 12.2 Eg.15
    Find the derivative of f(x)=x+1xf(x) = \frac{x + 1}{x}.
  12. 12.2 Eg.16
    Compute the derivative of sin⁡x\sin x.
  13. 12.2 Eg.17
    Compute the derivative of tan⁡x\tan x.
  14. 12.2 Eg.18
    Compute the derivative of f(x)=sin⁡2xf(x) = \sin^2 x.

Exercise 12.2

Practice · 27
  1. Ex 12.2 Q1
    Find the derivative of x2−2x^2 - 2 at x=10x = 10.
  2. Ex 12.2 Q2
    Find the derivative of xx at x=1x = 1.
  3. Ex 12.2 Q3
    Find the derivative of 99x99x at x=100x = 100.
  4. Find the derivative of the following functions from first principle.
    Ex 12.2 Q4(i)
    x3−27x^3 - 27
  5. Ex 12.2 Q4(ii)
    (x−1)(x−2)(x-1)(x-2)
  6. Ex 12.2 Q4(iii)
    1x2\frac{1}{x^2}
  7. Ex 12.2 Q4(iv)
    x+1x−1\frac{x+1}{x-1}
  8. Ex 12.2 Q5
    For the function f(x)=x100100+x9999+…+x22+x+1f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \ldots + \frac{x^2}{2} + x + 1. Prove that f′(1)=100f′(0)f'(1) = 100 f'(0).
  9. Ex 12.2 Q6
    Find the derivative of xn+axn−1+a2xn−2+…+an−1x+anx^n + ax^{n-1} + a^2 x^{n-2} + \ldots + a^{n-1}x + a^n for some fixed real number aa.
  10. For some constants aa and bb, find the derivative of
    Ex 12.2 Q7(i)
    (x−a)(x−b)(x-a)(x-b)
  11. Ex 12.2 Q7(ii)
    (ax2+b)2\left(ax^2 + b\right)^2
  12. Ex 12.2 Q7(iii)
    x−ax−b\frac{x-a}{x-b}
  13. Ex 12.2 Q8
    Find the derivative of xn−anx−a\frac{x^n - a^n}{x - a} for some constant aa.
  14. Find the derivative of
    Ex 12.2 Q9(i)
    2x−342x - \frac{3}{4}
  15. Ex 12.2 Q9(ii)
    (5x3+3x−1)(x−1)\left(5x^3 + 3x - 1\right)(x-1)
  16. Ex 12.2 Q9(iii)
    x−3(5+3x)x^{-3}(5 + 3x)
  17. Ex 12.2 Q9(iv)
    x5(3−6x−9)x^5\left(3 - 6x^{-9}\right)
  18. Ex 12.2 Q9(v)
    x−4(3−4x−5)x^{-4}\left(3 - 4x^{-5}\right)
  19. Ex 12.2 Q9(vi)
    2x+1−x23x−1\frac{2}{x+1} - \frac{x^2}{3x-1}
  20. Ex 12.2 Q10
    Find the derivative of cos⁡x\cos x from first principle.
  21. Find the derivative of the following functions:
    Ex 12.2 Q11(i)
    sin⁡xcos⁡x\sin x \cos x
  22. Ex 12.2 Q11(ii)
    sec⁡x\sec x
  23. Ex 12.2 Q11(iii)
    5sec⁡x+4cos⁡x5\sec x + 4\cos x
  24. Ex 12.2 Q11(iv)
    cosec⁡x\operatorname{cosec} x
  25. Ex 12.2 Q11(v)
    3cot⁡x+5cosec⁡x3\cot x + 5\operatorname{cosec} x
  26. Ex 12.2 Q11(vi)
    5sin⁡x−6cos⁡x+75\sin x - 6\cos x + 7
  27. Ex 12.2 Q11(vii)
    2tan⁡x−7sec⁡x2\tan x - 7\sec x

Miscellaneous Exercise on Chapter 12

41 q

Miscellaneous Examples

Worked · 8
  1. Find the derivative of ff from the first principle, where ff is given by
    Misc Eg.19(i)
    f(x)=2x+3x−2f(x) = \frac{2x+3}{x-2}
  2. Misc Eg.19(ii)
    f(x)=x+1xf(x) = x + \frac{1}{x}
  3. Find the derivative of f(x)f(x) from the first principle, where f(x)f(x) is
    Misc Eg.20(i)
    sin⁡x+cos⁡x\sin x + \cos x
  4. Misc Eg.20(ii)
    xsin⁡xx \sin x
  5. Compute derivative of
    Misc Eg.21(i)
    f(x)=sin⁡2xf(x) = \sin 2x
  6. Misc Eg.21(ii)
    g(x)=cot⁡xg(x) = \cot x
  7. Find the derivative of
    Misc Eg.22(i)
    x5−cos⁡xsin⁡x\frac{x^5 - \cos x}{\sin x}
  8. Misc Eg.22(ii)
    x+cos⁡xtan⁡x\frac{x + \cos x}{\tan x}

Miscellaneous Exercise

Practice · 33
  1. Find the derivative of the following functions from first principle:
    Misc Q1(i)
    −x-x
  2. Misc Q1(ii)
    (−x)−1(-x)^{-1}
  3. Misc Q1(iii)
    sin⁡(x+1)\sin(x+1)
  4. Misc Q1(iv)
    cos⁡(x−π8)\cos\left(x - \frac{\pi}{8}\right)
  5. Misc Q2
    Find the derivative of (x+a)(x + a), where aa is a fixed non-zero constant.
  6. Misc Q3
    Find the derivative of (px+q)(rx+s)(px + q)\left(\frac{r}{x} + s\right), where p,q,rp, q, r and ss are fixed non-zero constants.
  7. Misc Q4
    Find the derivative of (ax+b)(cx+d)2(ax + b)(cx + d)^2, where a,b,ca, b, c and dd are fixed non-zero constants.
  8. Misc Q5
    Find the derivative of ax+bcx+d\frac{ax + b}{cx + d}, where a,b,ca, b, c and dd are fixed non-zero constants.
  9. Misc Q6
    Find the derivative of 1+1x1−1x\frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}.
  10. Misc Q7
    Find the derivative of 1ax2+bx+c\frac{1}{ax^2 + bx + c}, where a,ba, b and cc are fixed non-zero constants.
  11. Misc Q8
    Find the derivative of ax+bpx2+qx+r\frac{ax + b}{px^2 + qx + r}, where a,b,p,qa, b, p, q and rr are fixed non-zero constants.
  12. Misc Q9
    Find the derivative of px2+qx+rax+b\frac{px^2 + qx + r}{ax + b}, where a,b,p,qa, b, p, q and rr are fixed non-zero constants.
  13. Misc Q10
    Find the derivative of ax4−bx2+cos⁡x\frac{a}{x^4} - \frac{b}{x^2} + \cos x, where aa and bb are fixed non-zero constants.
  14. Misc Q11
    Find the derivative of 4x−24\sqrt{x} - 2.
  15. Misc Q12
    Find the derivative of (ax+b)n(ax + b)^n, where aa and bb are fixed non-zero constants and nn is an integer.
  16. Misc Q13
    Find the derivative of (ax+b)n(cx+d)m(ax + b)^n (cx + d)^m, where a,b,ca, b, c and dd are fixed non-zero constants and mm and nn are integers.
  17. Misc Q14
    Find the derivative of sin⁡(x+a)\sin(x + a), where aa is a fixed non-zero constant.
  18. Misc Q15
    Find the derivative of cosec⁡xcot⁡x\operatorname{cosec} x \cot x.
  19. Misc Q16
    Find the derivative of cos⁡x1+sin⁡x\frac{\cos x}{1 + \sin x}.
  20. Misc Q17
    Find the derivative of sin⁡x+cos⁡xsin⁡x−cos⁡x\frac{\sin x + \cos x}{\sin x - \cos x}.
  21. Misc Q18
    Find the derivative of sec⁡x−1sec⁡x+1\frac{\sec x - 1}{\sec x + 1}.
  22. Misc Q19
    Find the derivative of sin⁡nx\sin^n x, where nn is an integer.
  23. Misc Q20
    Find the derivative of a+bsin⁡xc+dcos⁡x\frac{a + b\sin x}{c + d\cos x}, where a,b,ca, b, c and dd are fixed non-zero constants.
  24. Misc Q21
    Find the derivative of sin⁡(x+a)cos⁡x\frac{\sin(x + a)}{\cos x}, where aa is a fixed non-zero constant.
  25. Misc Q22
    Find the derivative of x4(5sin⁡x−3cos⁡x)x^4(5\sin x - 3\cos x).
  26. Misc Q23
    Find the derivative of (x2+1)cos⁡x(x^2 + 1)\cos x.
  27. Misc Q24
    Find the derivative of (ax2+sin⁡x)(p+qcos⁡x)(ax^2 + \sin x)(p + q\cos x), where a,pa, p and qq are fixed non-zero constants.
  28. Misc Q25
    Find the derivative of (x+cos⁡x)(x−tan⁡x)(x + \cos x)(x - \tan x).
  29. Misc Q26
    Find the derivative of 4x+5sin⁡x3x+7cos⁡x\frac{4x + 5\sin x}{3x + 7\cos x}.
  30. Misc Q27
    Find the derivative of x2cos⁡(π4)sin⁡x\frac{x^2 \cos\left(\frac{\pi}{4}\right)}{\sin x}.
  31. Misc Q28
    Find the derivative of x1+tan⁡x\frac{x}{1 + \tan x}.
  32. Misc Q29
    Find the derivative of (x+sec⁡x)(x−tan⁡x)(x + \sec x)(x - \tan x).
  33. Misc Q30
    Find the derivative of xsin⁡nx\frac{x}{\sin^n x}, where nn is an integer.