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Mathematics · 1st Puc Science

Ch 12Limits and Derivatives — 1st PUC Mathematics, concept-first.

Calculus is the branch of mathematics that studies how the value of a function changes as the points in its domain change. This chapter is a first introduction to calculus, and it is built up in careful stages rather than all at once.

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Limit Of Polynomial

A limit answers a simple question about a polynomial: as gets closer and closer to some number , what value does the polynomial settle near?

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Chapter contents

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12.1

Introduction

Calculus is the branch of mathematics that studies how the value of a function changes as the points in its domain change.

12.2

Intuitive Idea of Derivatives

Physical experiments show that a body dropped from a tall cliff covers a distance of metres in seconds. So the distance (in metres) as a function of time (in seconds) is:

12.3

Limits

The concept of a limit is the foundation of calculus. It answers a simple but powerful question: as the input to a function gets arbitrarily close to some number, what value does the output approach?…

12.3.1

Algebra of Limits

When you worked through the earlier illustrations in this chapter, you probably noticed something convenient: the process of taking a limit seemed to "play well" with addition, subtraction, multiplica…

12.3.2

Limits of Polynomials and Rational Functions

A polynomial function of degree is written as

+Worked Examplesi3 questions
  1. Example 1Find the limits: (i) $\lim_{x\to 1}\left[x^3 - x^2 + 1\right]$ (ii) $\lim_{x\to 3}\left[x(x + 1)\right]$ (iii) $\lim_{x\to -1}\left[1 + x +…Free
  2. Example 2Find the limits: (i) $\lim_{x\to 1}\left[\dfrac{x^2 + 1}{x + 100}\right]$ (ii) $\lim_{x\to 2}\left[\dfrac{x^3 - 4x^2 + 4x}{x^2 - 4}\right]$…Preview
  3. Example 3Evaluate: (i) $\lim_{x\to 1}\dfrac{x^{15} - 1}{x^{10} - 1}$ (ii) $\lim_{x\to 0}\dfrac{\sqrt{1 + x} - 1}{x}$Preview
12.4

Limits of Trigonometric Functions

33 Q

Before we can evaluate limits involving trigonometric functions, we need two general theorems about limits of functions.

+Worked Examplesi1 question
  1. Example 4Evaluate: (i) $\lim_{x\to 0}\dfrac{\sin 4x}{\sin 2x}$ (ii) $\lim_{x\to 0}\dfrac{\tan x}{x}$Preview
+Exercise 12.1i32 questions
  1. Q1$\lim_{x\to 3}(x + 3)$Free
  2. Q2$\lim_{x\to \pi}\left(x - \dfrac{22}{7}\right)$Free
  3. Q3$\lim_{r\to 1}\pi r^2$Free
  4. Q4$\lim_{x\to 4}\dfrac{4x + 3}{x - 2}$Preview
  5. Q5$\lim_{x\to -1}\dfrac{x^{10} + x^5 + 1}{x - 1}$Preview
  6. Q6$\lim_{x\to 0}\dfrac{(x + 1)^5 - 1}{x}$Preview
  7. Q7$\lim_{x\to 2}\dfrac{3x^2 - x - 10}{x^2 - 4}$Preview
  8. Q8$\lim_{x\to 3}\dfrac{x^4 - 81}{2x^2 - 5x - 3}$Preview
  9. Q9$\lim_{x\to 0}\dfrac{ax + b}{cx + 1}$Preview
  10. Q10$\lim_{z\to 1}\dfrac{z^{1/3} - 1}{z^{1/6} - 1}$Preview
  11. Q11$\lim_{x\to 1}\dfrac{ax^2 + bx + c}{cx^2 + bx + a},\ a + b + c \neq 0$Preview
  12. Q12$\lim_{x\to -2}\dfrac{\frac{1}{x} + \frac{1}{2}}{x + 2}$Preview
  13. Q13$\lim_{x\to 0}\dfrac{\sin ax}{bx}$Preview
  14. Q14$\lim_{x\to 0}\dfrac{\sin ax}{\sin bx},\ a, b \neq 0$Preview
  15. Q15$\lim_{x\to \pi}\dfrac{\sin(\pi - x)}{\pi(\pi - x)}$Preview
  16. Q16$\lim_{x\to 0}\dfrac{\cos x}{\pi - x}$Preview
  17. Q17$\lim_{x\to 0}\dfrac{\cos 2x - 1}{\cos x - 1}$Preview
  18. Q18$\lim_{x\to 0}\dfrac{ax + x\cos x}{b\sin x}$Preview
  19. Q19$\lim_{x\to 0} x\sec x$Preview
  20. Q20$\lim_{x\to 0}\dfrac{\sin ax + bx}{ax + \sin bx},\ a, b, a + b \neq 0$Preview
  21. Q21$\lim_{x\to 0}(\operatorname{cosec} x - \cot x)$Preview
  22. Q22$\lim_{x\to \frac{\pi}{2}}\dfrac{\tan 2x}{x - \frac{\pi}{2}}$Preview
  23. Q23Find $\lim_{x\to 0} f(x)$ and $\lim_{x\to 1} f(x)$, where $f(x) = \begin{cases} 2x + 3, & x \le 0 \\ 3(x + 1), & x > 0 \end{cases}$Preview
  24. Q24Find $\lim_{x\to 1} f(x)$, where $f(x) = \begin{cases} x^2 - 1, & x \le 1 \\ -x^2 - 1, & x > 1 \end{cases}$Preview
  25. Q25Evaluate $\lim_{x\to 0} f(x)$, where $f(x) = \begin{cases} \dfrac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$Preview
  26. Q26Find $\lim_{x\to 0} f(x)$, where $f(x) = \begin{cases} \dfrac{x}{|x|}, & x \neq 0 \\ 0, & x = 0 \end{cases}$Preview
  27. Q27Find $\lim_{x\to 5} f(x)$, where $f(x) = |x| - 5$.Preview
  28. Q28Suppose $f(x) = \begin{cases} a + bx, & x < 1 \\ 4, & x = 1 \\ b - ax, & x > 1 \end{cases}$ and if $\lim_{x\to 1} f(x) = f(1)$ what are poss…Preview
  29. Q29Let $a_1, a_2, \ldots, a_n$ be fixed real numbers and define a function $f(x) = (x - a_1)(x - a_2)\cdots(x - a_n)$. What is $\lim_{x\to a_1}…Preview
  30. Q30If $f(x) = \begin{cases} |x| + 1, & x < 0 \\ 0, & x = 0 \\ |x| - 1, & x > 0 \end{cases}$. For what value(s) of $a$ does $\lim_{x\to a} f(x)$…Preview
  31. Q31If the function $f(x)$ satisfies $\lim_{x\to 1}\dfrac{f(x) - 2}{x^2 - 1} = \pi$, evaluate $\lim_{x\to 1} f(x)$.Preview
  32. Q32If $f(x) = \begin{cases} mx^2 + n, & x < 0 \\ nx + m, & 0 \le x \le 1 \\ nx^3 + m, & x > 1 \end{cases}$. For what integers $m$ and $n$ does…Preview
12.5

Derivatives

The idea of a derivative grows directly out of a practical question: how fast is something changing? If you know where a car is at different times, you can work out its speed.

+Worked Examplesi8 questions
  1. Example 5Find the derivative at $x = 2$ of the function $f(x) = 3x$.Free
  2. Example 6Find the derivative of the function $f(x) = 2x^2 + 3x - 5$ at $x = -1$. Also prove that $f'(0) + 3f'(-1) = 0$.Free
  3. Example 7Find the derivative of $\sin x$ at $x = 0$.Free
  4. Example 8Find the derivative of $f(x) = 3$ at $x = 0$ and at $x = 3$.Preview
  5. Example 9Find the derivative of $f(x) = 10x$.Preview
  6. Example 10Find the derivative of $f(x) = x^2$.Preview
  7. Example 11Find the derivative of the constant function $f(x) = a$ for a fixed real number $a$.Preview
  8. Example 12Find the derivative of $f(x) = \dfrac{1}{x}$.Preview
12.5.1

Algebra of Derivative of Functions

Derivatives are defined through limits — the derivative of a function at a point is . Because limits themselves obey algebraic rules (the limit of a sum is the sum of the limits, and so on), it is nat…

12.5.2

Derivative of Polynomials and Trigonometric Functions

17 Q

The power of differentiation becomes truly useful when we apply it to the two most common families of functions: polynomials and trigonometric functions.

+Worked Examplesi6 questions
  1. Example 13Compute the derivative of $6x^{100} - x^{55} + x$.Free
  2. Example 14Find the derivative of $f(x) = 1 + x + x^2 + x^3 + \cdots + x^{50}$ at $x = 1$.Free
  3. Example 15Find the derivative of $f(x) = \dfrac{x + 1}{x}$.Preview
  4. Example 16Compute the derivative of $\sin x$.Preview
  5. Example 17Compute the derivative of $\tan x$.Preview
  6. Example 18Compute the derivative of $f(x) = \sin^2 x$.Preview
+Exercise 12.2i11 questions
  1. Q1Find the derivative of $x^2 - 2$ at $x = 10$.Free
  2. Q2Find the derivative of $x$ at $x = 1$.Free
  3. Q3Find the derivative of $99x$ at $x = 100$.Free
  4. Q4Find the derivative of the following functions from first principle. (i) $x^3 - 27$ (ii) $(x - 1)(x - 2)$ (iii) $\dfrac{1}{x^2}$ (iv) $\dfra…Preview
  5. Q5For the function $f(x) = \dfrac{x^{100}}{100} + \dfrac{x^{99}}{99} + \cdots + \dfrac{x^2}{2} + x + 1$. Prove that $f'(1) = 100\,f'(0)$.Preview
  6. Q6Find the derivative of $x^n + ax^{n-1} + a^2 x^{n-2} + \cdots + a^{n-1}x + a^n$ for some fixed real number $a$.Preview
  7. Q7For some constants $a$ and $b$, find the derivative of (i) $(x - a)(x - b)$ (ii) $(ax^2 + b)^2$ (iii) $\dfrac{x - a}{x - b}$Preview
  8. Q8Find the derivative of $\dfrac{x^n - a^n}{x - a}$ for some constant $a$.Preview
  9. Q9Find the derivative of (i) $2x - \dfrac{3}{4}$ (ii) $(5x^3 + 3x - 1)(x - 1)$ (iii) $x^{-3}(5 + 3x)$ (iv) $x^5(3 - 6x^{-9})$ (v) $x^{-4}(3 -…Preview
  10. Q10Find the derivative of $\cos x$ from first principle.Preview
  11. Q11Find the derivative of the following functions: (i) $\sin x\cos x$ (ii) $\sec x$ (iii) $5\sec x + 4\cos x$ (iv) $\operatorname{cosec} x$ (v)…Preview

Miscellaneous Examples

+Miscellaneous Examplesi4 questions
  1. Example 19Find the derivative of $f$ from the first principle, where $f$ is given by (i) $f(x) = \dfrac{2x + 3}{x - 2}$ (ii) $f(x) = x + \dfrac{1}{x}$Free
  2. Example 20Find the derivative of $f(x)$ from the first principle, where $f(x)$ is (i) $\sin x + \cos x$ (ii) $x\sin x$Free
  3. Example 21Compute the derivative of (i) $f(x) = \sin 2x$ (ii) $g(x) = \cot x$Preview
  4. Example 22Find the derivative of (i) $\dfrac{x^5 - \cos x}{\sin x}$ (ii) $\dfrac{x + \cos x}{\tan x}$Preview

Miscellaneous Exercise on Chapter 12

+Miscellaneous Exercisei30 questions
  1. Q1Find the derivative of the following functions from first principle: (i) $-x$ (ii) $(-x)^{-1}$ (iii) $\sin(x + 1)$ (iv) $\cos\left(x - \dfra…Free
  2. Q2Find the derivative of $(x + a)$.Free
  3. Q3Find the derivative of $(px + q)\left(\dfrac{r}{x} + s\right)$.Free
  4. Q4Find the derivative of $(ax + b)(cx + d)^2$.Preview
  5. Q5Find the derivative of $\dfrac{ax + b}{cx + d}$.Preview
  6. Q6Find the derivative of $\dfrac{1 + \frac{1}{x}}{1 - \frac{1}{x}}$.Preview
  7. Q7Find the derivative of $\dfrac{1}{ax^2 + bx + c}$.Preview
  8. Q8Find the derivative of $\dfrac{ax + b}{px^2 + qx + r}$.Preview
  9. Q9Find the derivative of $\dfrac{px^2 + qx + r}{ax + b}$.Preview
  10. Q10Find the derivative of $\dfrac{a}{x^4} - \dfrac{b}{x^2} + \cos x$.Preview
  11. Q11Find the derivative of $4\sqrt{x} - 2$.Preview
  12. Q12Find the derivative of $(ax + b)^n$.Preview
  13. Q13Find the derivative of $(ax + b)^n (cx + d)^m$.Preview
  14. Q14Find the derivative of $\sin(x + a)$.Preview
  15. Q15Find the derivative of $\operatorname{cosec} x\,\cot x$.Preview
  16. Q16Find the derivative of $\dfrac{\cos x}{1 + \sin x}$.Preview
  17. Q17Find the derivative of $\dfrac{\sin x + \cos x}{\sin x - \cos x}$.Preview
  18. Q18Find the derivative of $\dfrac{\sec x - 1}{\sec x + 1}$.Preview
  19. Q19Find the derivative of $\sin^n x$.Preview
  20. Q20Find the derivative of $\dfrac{a + b\sin x}{c + d\cos x}$.Preview
  21. Q21Find the derivative of $\dfrac{\sin(x + a)}{\cos x}$.Preview
  22. Q22Find the derivative of $x^4(5\sin x - 3\cos x)$.Preview
  23. Q23Find the derivative of $(x^2 + 1)\cos x$.Preview
  24. Q24Find the derivative of $(ax^2 + \sin x)(p + q\cos x)$.Preview
  25. Q25Find the derivative of $(x + \cos x)(x - \tan x)$.Preview
  26. Q26Find the derivative of $\dfrac{4x + 5\sin x}{3x + 7\cos x}$.Preview
  27. Q27Find the derivative of $\dfrac{x^2\cos\frac{\pi}{4}}{\sin x}$.Preview
  28. Q28Find the derivative of $\dfrac{x}{1 + \tan x}$.Preview
  29. Q29Find the derivative of $(x + \sec x)(x - \tan x)$.Preview
  30. Q30Find the derivative of $\dfrac{x}{\sin^n x}$.Preview

Summary

- Intuitive idea of a limit: means gets arbitrarily close to as gets arbitrarily close to (but ). The limit may exist even if is undefined. - Standard limits: - (for any rational ).

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 80 questions80 questions
  1. Q1Evaluate $\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3}$.Free
  2. Q2Evaluate $\lim_{x \to \frac{1}{2}} \dfrac{4x^2 - 1}{2x - 1}$.Free
  3. Q3Evaluate $\lim_{h \to 0} \dfrac{\sqrt{x + h} - \sqrt{x}}{h}$.Free
  4. Q4Evaluate $\lim_{x \to 0} \dfrac{(x + 2)^{\frac{1}{3}} - 2^{\frac{1}{3}}}{x}$.Preview
  5. Q5Evaluate $\lim_{x \to 1} \dfrac{(1 + x)^6 - 1}{(1 + x)^2 - 1}$.Preview
  6. Q6Evaluate $\lim_{x \to a} \dfrac{(2 + x)^{\frac{5}{2}} - (a + 2)^{\frac{5}{2}}}{x - a}$.Preview
  7. Q7Evaluate $\lim_{x \to 1} \dfrac{x^4 - \sqrt{x}}{\sqrt{x} - 1}$.Preview
  8. Q8Evaluate $\lim_{x \to 2} \dfrac{x^2 - 4}{\sqrt{3x - 2} - \sqrt{x + 2}}$.Preview
  9. Q9Evaluate $\lim_{x \to \sqrt{2}} \dfrac{x^4 - 4}{x^2 + 3\sqrt{2}\,x - 8}$.Preview
  10. Q10Evaluate $\lim_{x \to 1} \dfrac{x^7 - 2x^5 + 1}{x^3 - 3x^2 + 2}$.Preview
  11. Q11Evaluate $\lim_{x \to 0} \dfrac{\sqrt{1 + x^3} - \sqrt{1 - x^3}}{x^2}$.Preview
  12. Q12Evaluate $\lim_{x \to -3} \dfrac{x^3 + 27}{x^5 + 243}$.Preview
  13. Q13Evaluate $\lim_{x \to \frac{1}{2}} \left( \dfrac{8x - 3}{2x - 1} - \dfrac{4x^2 + 1}{4x^2 - 1} \right)$.Preview
  14. Q14Find '$n$', if $\lim_{x \to 2} \dfrac{x^n - 2^n}{x - 2} = 80$, $n \in \mathbf{N}$.Preview
  15. Q15Evaluate $\lim_{x \to a} \dfrac{\sin 3x}{\sin 7x}$.Preview
  16. Q16Evaluate $\lim_{x \to 0} \dfrac{\sin^2 2x}{\sin^2 4x}$.Preview
  17. Q17Evaluate $\lim_{x \to 0} \dfrac{1 - \cos 2x}{x^2}$.Preview
  18. Q18Evaluate $\lim_{x \to 0} \dfrac{2\sin x - \sin 2x}{x^3}$.Preview
  19. Q19Evaluate $\lim_{x \to 0} \dfrac{1 - \cos mx}{1 - \cos nx}$.Preview
  20. Q20Evaluate $\lim_{x \to \frac{\pi}{3}} \dfrac{\sqrt{1 - \cos 6x}}{\sqrt{2}\left(\frac{\pi}{3} - x\right)}$.Preview
  21. Q21Evaluate $\lim_{x \to \frac{\pi}{4}} \dfrac{\sin x - \cos x}{x - \frac{\pi}{4}}$.Preview
  22. Q22Evaluate $\lim_{x \to \frac{\pi}{6}} \dfrac{\sqrt{3}\,\sin x - \cos x}{x - \frac{\pi}{6}}$.Preview
  23. Q23Evaluate $\lim_{x \to 0} \dfrac{\sin 2x + 3x}{2x + \tan 3x}$.Preview
  24. Q24Evaluate $\lim_{x \to a} \dfrac{\sin x - \sin a}{\sqrt{x} - \sqrt{a}}$.Preview
  25. Q25Evaluate $\lim_{x \to \frac{\pi}{6}} \dfrac{\cot^2 x - 3}{\operatorname{cosec} x - 2}$.Preview
  26. Q26Evaluate $\lim_{x \to 0} \dfrac{\sqrt{2} - \sqrt{1 + \cos x}}{\sin^2 x}$.Preview
  27. Q27Evaluate $\lim_{x \to 0} \dfrac{\sin x - 2\sin 3x + \sin 5x}{x}$.Preview
  28. Q28If $\lim_{x \to 1} \dfrac{x^4 - 1}{x - 1} = \lim_{x \to k} \dfrac{x^3 - k^3}{x^2 - k^2}$, then find the value of $k$.Preview
  29. Q29Differentiate with respect to $x$: $\dfrac{x^4 + x^3 + x^2 + 1}{x}$.Preview
  30. Q30Differentiate with respect to $x$: $\left(x + \dfrac{1}{x}\right)^3$.Preview
  31. Q31Differentiate with respect to $x$: $(3x + 5)(1 + \tan x)$.Preview
  32. Q32Differentiate with respect to $x$: $(\sec x - 1)(\sec x + 1)$.Preview
  33. Q33Differentiate with respect to $x$: $\dfrac{3x + 4}{5x^2 - 7x + 9}$.Preview
  34. Q34Differentiate with respect to $x$: $\dfrac{x^5 - \cos x}{\sin x}$.Preview
  35. Q35Differentiate with respect to $x$: $\dfrac{x^2 \cos \frac{\pi}{4}}{\sin x}$.Preview
  36. Q36Differentiate with respect to $x$: $(ax^2 + \cot x)(p + q\cos x)$.Preview
  37. Q37Differentiate with respect to $x$: $\dfrac{a + b\sin x}{c + d\cos x}$.Preview
  38. Q38Differentiate with respect to $x$: $(\sin x + \cos x)^2$.Preview
  39. Q39Differentiate with respect to $x$: $(2x - 7)^2 (3x + 5)^3$.Preview
  40. Q40Differentiate with respect to $x$: $x^2 \sin x + \cos 2x$.Preview
  41. Q41Differentiate with respect to $x$: $\sin^3 x \cos^3 x$.Preview
  42. Q42Differentiate with respect to $x$: $\dfrac{1}{ax^2 + bx + c}$.Preview
  43. Q43Differentiate with respect to $x$ using first principle: $\cos(x^2 + 1)$.Preview
  44. Q44Differentiate with respect to $x$ using first principle: $\dfrac{ax + b}{cx + d}$.Preview
  45. Q45Differentiate with respect to $x$ using first principle: $x^{\frac{2}{3}}$.Preview
  46. Q46Differentiate with respect to $x$ using first principle: $x\cos x$.Preview
  47. Q47Evaluate $\lim_{y \to 0} \dfrac{(x + y)\sec(x + y) - x\sec x}{y}$.Preview
  48. Q48Evaluate $\lim_{x \to 0} \dfrac{\sin(\alpha + \beta)x + \sin(\alpha - \beta)x + \sin 2\alpha x}{\cos 2\beta x - \cos 2\alpha x} \cdot x$.Preview
  49. Q49Evaluate $\lim_{x \to \frac{\pi}{4}} \dfrac{\tan^3 x - \tan x}{\cos\left(x + \frac{\pi}{4}\right)}$.Preview
  50. Q50Evaluate $\lim_{x \to \pi} \dfrac{1 - \sin\frac{x}{2}}{\cos\frac{x}{2}\left(\cos\frac{x}{4} - \sin\frac{x}{4}\right)}$.Preview
  51. Q51Show that $\lim_{x \to 4} \dfrac{|x - 4|}{x - 4}$ does not exists.Preview
  52. Q52Let $f(x) = \begin{cases} \dfrac{k\cos x}{\pi - 2x} & \text{when } x \neq \frac{\pi}{2} \\ 3 & x = \frac{\pi}{2} \end{cases}$ and if $\lim_{…Preview
  53. Q53Let $f(x) = \begin{cases} x + 2 & x \leq 1 \\ cx^2 & x > -1 \end{cases}$, find '$c$' if $\lim_{x \to -1} f(x)$ exists.Preview
  54. Q54$\lim_{x \to \pi} \dfrac{\sin x}{x - \pi}$ is (A) $1$ (B) $2$ (C) $-1$ (D) $-2$Preview
  55. Q55$\lim_{x \to 0} \dfrac{x^2 \cos x}{1 - \cos x}$ is (A) $2$ (B) $\dfrac{3}{2}$ (C) $\dfrac{-3}{2}$ (D) $1$Preview
  56. Q56$\lim_{x \to 0} \dfrac{(1 + x)^n - 1}{x}$ is (A) $n$ (B) $1$ (C) $-n$ (D) $0$Preview
  57. Q57$\lim_{x \to 1} \dfrac{x^m - 1}{x^n - 1}$ is (A) $1$ (B) $\dfrac{m}{n}$ (C) $-\dfrac{m}{n}$ (D) $\dfrac{m^2}{n^2}$Preview
  58. Q58$\lim_{x \to 0} \dfrac{1 - \cos 4\theta}{1 - \cos 6\theta}$ is (A) $\dfrac{4}{9}$ (B) $\dfrac{1}{2}$ (C) $\dfrac{-1}{2}$ (D) $-1$Preview
  59. Q59$\lim_{x \to 0} \dfrac{\operatorname{cosec} x - \cot x}{x}$ is (A) $\dfrac{-1}{2}$ (B) $1$ (C) $\dfrac{1}{2}$ (D) $1$Preview
  60. Q60$\lim_{x \to 0} \dfrac{\sin x}{\sqrt{x + 1} - \sqrt{1 - x}}$ is (A) $2$ (B) $0$ (C) $1$ (D) $-1$Preview
  61. Q61$\lim_{x \to \frac{\pi}{4}} \dfrac{\sec^2 x - 2}{\tan x - 1}$ is (A) $3$ (B) $1$ (C) $0$ (D) $\sqrt{2}$Preview
  62. Q62$\lim_{x \to 1} \dfrac{\left(\sqrt{x} - 1\right)(2x - 3)}{2x^2 + x - 3}$ is (A) $\dfrac{1}{10}$ (B) $\dfrac{-1}{10}$ (C) $1$ (D) None of the…Preview
  63. Q63If $f(x) = \begin{cases} \dfrac{\sin[x]}{[x]}, & [x] \neq 0 \\ 0, & [x] = 0 \end{cases}$, where $[.]$ denotes the greatest integer function,…Preview
  64. Q64$\lim_{x \to 0} \dfrac{|\sin x|}{x}$ is (A) $1$ (B) $-1$ (C) does not exist (D) None of thesePreview
  65. Q65Let $f(x) = \begin{cases} x^2 - 1, & 0 < x < 2 \\ 2x + 3, & 2 \leq x < 3 \end{cases}$, the quadratic equation whose roots are $\lim_{x \to 2…Preview
  66. Q66$\lim_{x \to 0} \dfrac{\tan 2x - x}{3x - \sin x}$ is (A) $2$ (B) $\dfrac{1}{2}$ (C) $\dfrac{-1}{2}$ (D) $\dfrac{1}{4}$Preview
  67. Q67Let $f(x) = x - [x]; \in \mathbf{R}$, then $f'\left(\dfrac{1}{2}\right)$ is (A) $\dfrac{3}{2}$ (B) $1$ (C) $0$ (D) $-1$Preview
  68. Q68If $y = \sqrt{x} + \dfrac{1}{\sqrt{x}}$, then $\dfrac{dy}{dx}$ at $x = 1$ is (A) $1$ (B) $\dfrac{1}{2}$ (C) $\dfrac{1}{\sqrt{2}}$ (D) $0$Preview
  69. Q69If $f(x) = \dfrac{x - 4}{2\sqrt{x}}$, then $f'(1)$ is (A) $\dfrac{5}{4}$ (B) $\dfrac{4}{5}$ (C) $1$ (D) $0$Preview
  70. Q70If $y = \dfrac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}$, then $\dfrac{dy}{dx}$ is (A) $\dfrac{-4x}{(x^2 - 1)^2}$ (B) $\dfrac{-4x}{x^2 - 1}$ (C…Preview
  71. Q71If $y = \dfrac{\sin x + \cos x}{\sin x - \cos x}$, then $\dfrac{dy}{dx}$ at $x = 0$ is (A) $-2$ (B) $0$ (C) $\dfrac{1}{2}$ (D) does not exis…Preview
  72. Q72If $y = \dfrac{\sin(x + 9)}{\cos x}$ then $\dfrac{dy}{dx}$ at $x = 0$ is (A) $\cos 9$ (B) $\sin 9$ (C) $0$ (D) $1$Preview
  73. Q73If $f(x) = 1 + x + \dfrac{x^2}{2} + ... + \dfrac{x^{100}}{100}$, then $f'(1)$ is equal to (A) $\dfrac{1}{100}$ (B) $100$ (C) does not exist…Preview
  74. Q74If $f(x) = \dfrac{x^n - a^n}{x - a}$ for some constant '$a$', then $f'(a)$ is (A) $1$ (B) $0$ (C) does not exist (D) $\dfrac{1}{2}$Preview
  75. Q75If $f(x) = x^{100} + x^{99} + ... + x + 1$, then $f'(1)$ is equal to (A) $5050$ (B) $5049$ (C) $5051$ (D) $50051$Preview
  76. Q76If $f(x) = 1 - x + x^2 - x^3 ... - x^{99} + x^{100}$, then $f'(1)$ is euqal to (A) $150$ (B) $-50$ (C) $-150$ (D) $50$Preview
  77. Q77If $f(x) = \dfrac{\tan x}{x - \pi}$, then $\lim_{x \to \pi} f(x) = $ ________.Preview
  78. Q78$\lim_{x \to 0} \left( \sin mx \cot \dfrac{x}{\sqrt{3}} \right) = 2$, then $m = $ ________.Preview
  79. Q79If $y = 1 + \dfrac{x}{1!} + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + ...$, then $\dfrac{dy}{dx} = $ ________.Preview
  80. Q80$\lim_{x \to 3^+} \dfrac{x}{[x]} = $ ________.Preview