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

Find the limits:
12.1 Eg.1(i) $\lim_{x \to 1}\left[x^3 - x^2 + 1\right]$
12.1 Eg.1(ii) $\lim_{x \to 3}\left[x(x+1)\right]$
12.1 Eg.1(iii) $\lim_{x \to -1}\left[1 + x + x^2 + \ldots + x^{10}\right]$
Find the limits:
12.1 Eg.2(i) $\lim_{x \to 1}\left[\frac{x^2 + 1}{x + 100}\right]$
12.1 Eg.2(ii) $\lim_{x \to 2}\left[\frac{x^3 - 4x^2 + 4x}{x^2 - 4}\right]$
12.1 Eg.2(iii) $\lim_{x \to 2}\left[\frac{x^2 - 4}{x^3 - 4x^2 + 4x}\right]$
12.1 Eg.2(iv) $\lim_{x \to 2}\left[\frac{x^3 - 2x^2}{x^2 - 5x + 6}\right]$
12.1 Eg.2(v) $\lim_{x \to 1}\left[\frac{x-2}{x^2 - x} - \frac{1}{x^3 - 3x^2 + 2x}\right]$
Evaluate:
12.1 Eg.3(i) $\lim_{x \to 1}\frac{x^{15} - 1}{x^{10} - 1}$
12.1 Eg.3(ii) $\lim_{x \to 0}\frac{\sqrt{1+x} - 1}{x}$
Evaluate:
12.1 Eg.4(i) $\lim_{x \to 0}\frac{\sin 4x}{\sin 2x}$
12.1 Eg.4(ii) $\lim_{x \to 0}\frac{\tan x}{x}$

Exercise 12.1

Practice · 32

Ex 12.1 Q1 $\lim_{x \to 3} x + 3$
Ex 12.1 Q2 $\lim_{x \to \pi} \left( x - \frac{22}{7} \right)$
Ex 12.1 Q3 $\lim_{r \to 1} \pi r^2$
Ex 12.1 Q4 $\lim_{x \to 4} \frac{4x + 3}{x - 2}$
Ex 12.1 Q5 $\lim_{x \to -1} \frac{x^{10} + x^5 + 1}{x - 1}$
Ex 12.1 Q6 $\lim_{x \to 0} \frac{(x + 1)^5 - 1}{x}$
Ex 12.1 Q7 $\lim_{x \to 2} \frac{3x^2 - x - 10}{x^2 - 4}$
Ex 12.1 Q8 $\lim_{x \to 3} \frac{x^4 - 81}{2x^2 - 5x - 3}$
Ex 12.1 Q9 $\lim_{x \to 0} \frac{ax + b}{cx + 1}$
Ex 12.1 Q10 $\lim_{z \to 1} \frac{z^{\frac{1}{3}} - 1}{z^{\frac{1}{6}} - 1}$
Ex 12.1 Q11 $\lim_{x \to 1} \frac{ax^2 + bx + c}{cx^2 + bx + a}$
Ex 12.1 Q12 $\lim_{x \to -2} \dfrac{\frac{1}{x} + \frac{1}{2}}{x + 2}$
Ex 12.1 Q13 $\lim_{x \to 0} \frac{\sin ax}{bx}$
Ex 12.1 Q14 $\lim_{x \to 0} \frac{\sin ax}{\sin bx}$
Ex 12.1 Q15 $\lim_{x \to \pi} \frac{\sin(\pi - x)}{\pi(\pi - x)}$
Ex 12.1 Q16 $\lim_{x \to 0} \frac{\cos x}{\pi - x}$
Ex 12.1 Q17 $\lim_{x \to 0} \frac{\cos 2x - 1}{\cos x - 1}$
Ex 12.1 Q18 $\lim_{x \to 0} \frac{ax + x \cos x}{b \sin x}$
Ex 12.1 Q19 $\lim_{x \to 0} x \sec x$
Ex 12.1 Q20 $\lim_{x \to 0} \frac{\sin ax + bx}{ax + \sin bx}$
Ex 12.1 Q21 $\lim_{x \to 0} (\operatorname{cosec} x - \cot x)$
Ex 12.1 Q22 $\lim_{x \to \frac{\pi}{2}} \dfrac{\tan 2x}{x - \dfrac{\pi}{2}}$
Ex 12.1 Q23 $\lim_{x \to 0} f(x)$
Ex 12.1 Q24 $\lim_{x \to 1} f(x)$
Ex 12.1 Q25 $\lim_{x \to 0} f(x)$
Ex 12.1 Q26 $\lim_{x \to 0} f(x)$
Ex 12.1 Q27 $\lim_{x \to 5} f(x)$
Ex 12.1 Q28 $f(x) = \begin{cases} a + bx, & x < 1 \\ 4, & x = 1 \\ b - ax, & x > 1 \end{cases}$
Ex 12.1 Q29 $a_1, a_2, \ldots, a_n$
Ex 12.1 Q30 $f(x) = \begin{cases} |x| + 1, & x < 0 \\ 0, & x = 0 \\ |x| - 1, & x > 0 \end{cases}$
Ex 12.1 Q31 $f(x)$
Ex 12.1 Q32 $f(x) = \begin{cases} mx^2 + n, & x < 0 \\ nx + m, & 0 \le x \le 1 \\ nx^3 + m, & x > 1 \end{cases}$

12.5 Derivatives

41 q

Solved Examples

Worked · 14

12.2 Eg.5 $x = 2$
12.2 Eg.6 $f(x) = 2x^2 + 3x - 5$
12.2 Eg.7 $\sin x$
12.2 Eg.8 $f(x) = 3$
12.2 Eg.9 $f(x) = 10x$
12.2 Eg.10 $f(x) = x^2$
12.2 Eg.11 $f(x) = a$
12.2 Eg.12 $f(x) = \frac{1}{x}$
12.2 Eg.13 $6x^{100} - x^{55} + x$
12.2 Eg.14 $f(x) = 1 + x + x^2 + x^3 + \ldots + x^{50}$
12.2 Eg.15 $f(x) = \frac{x + 1}{x}$
12.2 Eg.16 $\sin x$
12.2 Eg.17 $\tan x$
12.2 Eg.18 $f(x) = \sin^2 x$

Exercise 12.2

Practice · 27

Ex 12.2 Q1 $x^2 - 2$
Ex 12.2 Q2 $x$
Ex 12.2 Q3 $99x$
Find the derivative of the following functions from first principle.
Ex 12.2 Q4(i) $x^3 - 27$
Ex 12.2 Q4(ii) $(x-1)(x-2)$
Ex 12.2 Q4(iii) $\frac{1}{x^2}$
Ex 12.2 Q4(iv) $\frac{x+1}{x-1}$
Ex 12.2 Q5 $f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \ldots + \frac{x^2}{2} + x + 1$
Ex 12.2 Q6 $x^n + ax^{n-1} + a^2 x^{n-2} + \ldots + a^{n-1}x + a^n$
For some constants $a$ and $b$ , find the derivative of
Ex 12.2 Q7(i) $(x-a)(x-b)$
Ex 12.2 Q7(ii) $\left(ax^2 + b\right)^2$
Ex 12.2 Q7(iii) $\frac{x-a}{x-b}$
Ex 12.2 Q8 $\frac{x^n - a^n}{x - a}$
Find the derivative of
Ex 12.2 Q9(i) $2x - \frac{3}{4}$
Ex 12.2 Q9(ii) $\left(5x^3 + 3x - 1\right)(x-1)$
Ex 12.2 Q9(iii) $x^{-3}(5 + 3x)$
Ex 12.2 Q9(iv) $x^5\left(3 - 6x^{-9}\right)$
Ex 12.2 Q9(v) $x^{-4}\left(3 - 4x^{-5}\right)$
Ex 12.2 Q9(vi) $\frac{2}{x+1} - \frac{x^2}{3x-1}$
Ex 12.2 Q10 $\cos x$
Find the derivative of the following functions:
Ex 12.2 Q11(i) $\sin x \cos x$
Ex 12.2 Q11(ii) $\sec x$
Ex 12.2 Q11(iii) $5\sec x + 4\cos x$
Ex 12.2 Q11(iv) $\operatorname{cosec} x$
Ex 12.2 Q11(v) $3\cot x + 5\operatorname{cosec} x$
Ex 12.2 Q11(vi) $5\sin x - 6\cos x + 7$
Ex 12.2 Q11(vii) $2\tan x - 7\sec x$

Miscellaneous Exercise on Chapter 12

41 q

Miscellaneous Examples

Worked · 8

Find the derivative of $f$ from the first principle, where $f$ is given by
Misc Eg.19(i) $f(x) = \frac{2x+3}{x-2}$
Misc Eg.19(ii) $f(x) = x + \frac{1}{x}$
Find the derivative of $f(x)$ from the first principle, where $f(x)$ is
Misc Eg.20(i) $\sin x + \cos x$
Misc Eg.20(ii) $x \sin x$
Compute derivative of
Misc Eg.21(i) $f(x) = \sin 2x$
Misc Eg.21(ii) $g(x) = \cot x$
Find the derivative of
Misc Eg.22(i) $\frac{x^5 - \cos x}{\sin x}$
Misc Eg.22(ii) $\frac{x + \cos x}{\tan x}$

Miscellaneous Exercise

Practice · 33

Find the derivative of the following functions from first principle:
Misc Q1(i) $-x$
Misc Q1(ii) $(-x)^{-1}$
Misc Q1(iii) $\sin(x+1)$
Misc Q1(iv) $\cos\left(x - \frac{\pi}{8}\right)$
Misc Q2 $(x + a)$
Misc Q3 $(px + q)\left(\frac{r}{x} + s\right)$
Misc Q4 $(ax + b)(cx + d)^2$
Misc Q5 $\frac{ax + b}{cx + d}$
Misc Q6 $\frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}$
Misc Q7 $\frac{1}{ax^2 + bx + c}$
Misc Q8 $\frac{ax + b}{px^2 + qx + r}$
Misc Q9 $\frac{px^2 + qx + r}{ax + b}$
Misc Q10 $\frac{a}{x^4} - \frac{b}{x^2} + \cos x$
Misc Q11 $4\sqrt{x} - 2$
Misc Q12 $(ax + b)^n$
Misc Q13 $(ax + b)^n (cx + d)^m$
Misc Q14 $\sin(x + a)$
Misc Q15 $\operatorname{cosec} x \cot x$
Misc Q16 $\frac{\cos x}{1 + \sin x}$
Misc Q17 $\frac{\sin x + \cos x}{\sin x - \cos x}$
Misc Q18 $\frac{\sec x - 1}{\sec x + 1}$
Misc Q19 $\sin^n x$
Misc Q20 $\frac{a + b\sin x}{c + d\cos x}$
Misc Q21 $\frac{\sin(x + a)}{\cos x}$
Misc Q22 $x^4(5\sin x - 3\cos x)$
Misc Q23 $(x^2 + 1)\cos x$
Misc Q24 $(ax^2 + \sin x)(p + q\cos x)$
Misc Q25 $(x + \cos x)(x - \tan x)$
Misc Q26 $\frac{4x + 5\sin x}{3x + 7\cos x}$
Misc Q27 $\frac{x^2 \cos\left(\frac{\pi}{4}\right)}{\sin x}$
Misc Q28 $\frac{x}{1 + \tan x}$
Misc Q29 $(x + \sec x)(x - \tan x)$
Misc Q30 $\frac{x}{\sin^n x}$