Limits and Derivatives

1. Check this before you revise anything

The Chain Rule is not part of the current book — but one of its special cases still is. CBSE's formative-only block lists "Derivatives of composite functions (Chain rule)" for this chapter, and the current book never states, proves, or names a chain rule anywhere. But the Miscellaneous Exercise's own Q12 and Q13 still ask you to differentiate $(ax+b{)}^n$ and $(ax+b{)}^n(cx+d{)}^m$.

The book's own route to these is not "apply the chain rule" — it's a direct first-principles substitution using the standard algebraic limit already proved earlier in the chapter (Section 4 below). Section 8 shows exactly how, since this is a genuinely examinable technique despite the general chain rule staying out of scope.

Limits of exponential and logarithmic functions are named in the syllabus but never appear in this book. The syllabus line for this chapter explicitly reads "Limits of polynomials and rational functions trigonometric, exponential and logarithmic functions" — but the current 2026-27 book has no exponential or logarithmic content anywhere (confirmed by a full-text search for "exponential", "logarithm", and "$e^x$" — zero hits).

Since this is named as summative, not formative-only, Section 5 below teaches the two standard limits properly, clearly marked as syllabus-sourced rather than drawn from this book.

Derivatives of $e^x$ and $\log x$, by contrast, are not part of this chapter's syllabus at all. The syllabus explicitly restricts the derivative portion of this chapter to "polynomial and trigonometric functions" — exponential and logarithmic derivatives belong to Class 12. Don't expect them tested from this chapter.

2. What this chapter covers

3. Limits — the intuitive idea

For a function $f$ and a point $a$, ${\lim }_{x\to a}f(x)$ is the value $f(x)$ is expected to take as $x$ gets arbitrarily close to $a$ — which may or may not be the same as $f(a)$ itself, and may exist even when $f(a)$ doesn't.

Since $x$ can approach $a$ from below or above, there are two one-sided limits: the left-hand limit ${\lim }_{x\to a^{-}}f(x)$ (dictated by values just left of $a$) and the right-hand limit ${\lim }_{x\to a^{+}}f(x)$ (dictated by values just right of $a$). The limit ${\lim }_{x\to a}f(x)$ exists, and equals their common value, exactly when the two one-sided limits agree.

Worked, mirroring the textbook's own Illustration 9. For $f(x)=x-2$ when $x<0$, $f(0)=0$, and $f(x)=x+2$ when $x>0$: the left-hand limit at $0$ is $0-2=-2$, and the right-hand limit is $0+2=2$. Since these disagree, ${\lim }_{x\to 0}f(x)$ does not exist — even though $f(0)=0$ is perfectly well defined.

4. Algebra of limits and the standard algebraic limit

If ${\lim }_{x\to a}f(x)$ and ${\lim }_{x\to a}g(x)$ both exist, limits respect sums, differences, products, and quotients (denominator nonzero) termwise: $\lim [f\pm g]=\lim f\pm \lim g$, $\lim [fg]=\lim f\cdot \lim g$, $\lim [f/g]=\lim f/\lim g$. Since ${\lim }_{x\to a}x=a$, this immediately gives ${\lim }_{x\to a}P(x)=P(a)$ for any polynomial $P$, and the same by direct substitution for a rational function whenever the denominator doesn't vanish at $a$.

When substitution gives $0/0$, factor out and cancel the term causing both to vanish. The standard tool for this is:

${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

proved by factoring $x^n-a^n=(x-a)(x^{n-1}+x^{n-2}a+\cdots +a^{n-1})$ and cancelling the $(x-a)$ — this holds for any positive integer $n$, and (stated without proof here) for any rational $n$ as well, provided $a>0$.

Worked, mirroring the textbook's own Example 3(i). Evaluate ${\lim }_{x\to 1}\frac{x^{15}-1}{x^{10}-1}$. Write as $\frac{x^{15}-1}{x-1}÷\frac{x^{10}-1}{x-1}$, so the limit is $15(1{)}^{14}÷10(1{)}^9=15/10=3/2$.

5. Limits of trigonometric, exponential, and logarithmic functions

The Sandwich Theorem: if $f(x)\le g(x)\le h(x)$ near $a$ and $f,h$ share the same limit $l$ at $a$, then $g$ also has limit $l$ at $a$. The book proves the inequality $\sin x<x<\tan x$ for $0<x<\pi /2$ geometrically (comparing the areas of a triangle, a sector, and a larger triangle in a unit circle), which rearranges to $\cos x<\frac{\sin x}{x}<1$. Since $\cos x\to 1$ as $x\to 0$, sandwiching gives the two standard trigonometric limits:

${\lim }_{x\to 0}\frac{\sin x}{x}=1,{\lim }_{x\to 0}\frac{1-\cos x}{x}=0$

the second following from the first via the identity $1-\cos x=2{\sin }^2(x/2)$.

Worked, mirroring the textbook's own Example 4(i). Evaluate ${\lim }_{x\to 0}\frac{\sin 4x}{\sin 2x}$. Write as $2\cdot \frac{\sin 4x}{4x}÷\frac{\sin 2x}{2x}$, so the limit is $2\cdot 1÷1=2$ (since $4x\to 0$ and $2x\to 0$ together with $x\to 0$).

Limits of exponential and logarithmic functions, named in the syllabus but not derived in this book. Two standard limits — consequences of how the number $e$ is itself defined — are worth knowing for this chapter even though the current book never states them:

${\lim }_{x\to 0}\frac{e^x-1}{x}=1,{\lim }_{x\to 0}\frac{\log (1+x)}{x}=1$

Both are usually just quoted at this level, since a full derivation needs either the series definition of $e^x$ or L'Hôpital's Rule, neither available yet.

What matters here is recognising and using them, the same way $\sin x/x\to 1$ is used above.

6. Derivatives — definition from first principles

Definition. For a function $f$ and a point $a$ in its domain, the derivative of $f$ at $a$ is

$f^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$

provided the limit exists. Doing this at a general point $x$ instead of a fixed $a$ defines the derivative function $f^{\prime }(x)$, also written $\frac{df}{dx}$ or $\frac{dy}{dx}$.

Geometric meaning. For nearby points $P=(a,f(a))$ and $Q=(a+h,f(a+h))$ on the graph of $f$, the ratio inside the limit is exactly the slope of the chord $PQ$. As $h\to 0$, $Q$ slides along the curve toward $P$ and the chord's slope approaches the slope of the tangent to the curve at $P$.

So $f^{\prime }(a)$ is the tangent's slope at $x=a$ — this is exactly the "instantaneous velocity" idea the chapter opens with: a body's position $s=4.9t^2$ has instantaneous velocity at $t=2$ equal to the slope of the tangent to that curve at $t=2$.

Worked, mirroring the textbook's own Example 6. Find $f^{\prime }(-1)$ for $f(x)=2x^2+3x-5$. $f^{\prime }(-1)={\lim }_{h\to 0}\frac{f(-1+h)-f(-1)}{h}$. Expanding $f(-1+h)=2(-1+h{)}^2+3(-1+h)-5=2h^2-4h+2-3+3h-5=2h^2-h-6$, and $f(-1)=2-3-5=-6$, so the difference is $2h^2-h$, giving ${\lim }_{h\to 0}(2h-1)=-1$.

7. Algebra of derivatives

If $f^{\prime }$ and $g^{\prime }$ both exist, then $(f\pm g{)}^{\prime }=f^{\prime }\pm g^{\prime }$, and — writing $u=f(x)$, $v=g(x)$ for brevity —

$\text{Product rule: }(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime },\text{Quotient rule: }{\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$

Derivative of $x^n$. From first principles, $\frac{d}{dx}(x^n)=n{x}^{n-1}$ for any positive integer $n$ — this is exactly the algebraic limit from Section 4 applied to $f(x+h)-f(x)$ over $h$. Combined with the sum rule, the derivative of any polynomial follows termwise: for $f(x)=a_0+a_{1}x+\cdots +a_n{x}^n$, $f^{\prime }(x)=a_1+2a_{2}x+\cdots +n{a}_n{x}^{n-1}$.

Worked, mirroring the textbook's own Example 13. Differentiate $6x^{100}-x^{55}+x$. Termwise: $600x^{99}-55x^{54}+1$.

8. Differentiating $(ax+b{)}^n$ — the genuinely examinable case beyond a plain power

Neither the general chain rule nor a named formula for this is in the book — but the technique is a direct extension of the first-principles definition, and it's needed for the Miscellaneous Exercise's own Q12 and Q13.

Let $f(x)=(ax+b{)}^n$ and write $u=ax+b$. Then $f(x+h)-f(x)=(a(x+h)+b{)}^n-(ax+b{)}^n=(u+ah{)}^n-u^n$. Dividing by $h$ and writing $\Delta u=ah$:

$\frac{(u+ah{)}^n-u^n}{h}=a\cdot \frac{(u+\Delta u{)}^n-u^n}{\Delta u}$

As $h\to 0$, $\Delta u\to 0$ too, and the fraction on the right is exactly the standard algebraic limit from Section 4, tending to $n{u}^{n-1}$. So:

$\frac{d}{dx}(ax+b{)}^n=an(ax+b{)}^{n-1}$

For a product like $(ax+b{)}^n(cx+d{)}^m$, apply the product rule with each factor's derivative found the same way.

9. Derivatives of trigonometric functions

Only $\sin x$ and $\cos x$ get their own first-principles derivation and appear in the book's own final Summary as standalone results to know — $\tan x$, $\cot x$, $\sec x$, and $\csc x$ are meant to be produced from these two via the quotient rule, not memorised separately, though Exercise 12.2's own Q11 does test all of them directly.

Derivative of $\sin x$, mirroring the textbook's own Example 16. Using the compound-angle identity $\sin (x+h)-\sin x=2\cos \left(x+\frac{h}{2}\right)\sin \left(\frac{h}{2}\right)$:

$\frac{d}{dx}(\sin x)={\lim }_{h\to 0}\frac{2\cos (x+h/2)\sin (h/2)}{h}={\lim }_{h\to 0}\cos \left(x+\frac{h}{2}\right)\cdot \frac{\sin (h/2)}{h/2}=\cos x$

using the standard limit $\sin \theta /\theta \to 1$ (Section 5) with $\theta =h/2\to 0$. The same method (compound-angle identity for $\cos (x+h)-\cos x$) gives $\frac{d}{dx}(\cos x)=-\sin x$ — this is exactly what Exercise 12.2's own Q10 asks you to reproduce.

Everything else follows from the quotient rule. For $\tan x=\sin x/\cos x$, mirroring the textbook's own Example 17: $u=\sin x,u^{\prime }=\cos x$; $v=\cos x,v^{\prime }=-\sin x$, giving $\frac{d}{dx}(\tan x)=\frac{\cos x\cos x-\sin x(-\sin x)}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}={\sec }^{2}x$. The same approach on $\cot x=\cos x/\sin x$, $\sec x=1/\cos x$, and $\csc x=1/\sin x$ gives $-{\csc }^{2}x$, $\sec x\tan x$, and $-\csc x\cot x$ respectively.

Summary

• ${\lim }_{x\to a}f(x)$ exists exactly when the left-hand and right-hand limits agree; it need not equal $f(a)$, and may exist even where $f(a)$ doesn't.
• Algebra of limits: limits distribute over $\pm$, $\times$, $÷$ (nonzero denominator) whenever the individual limits exist; polynomials and rational functions (denominator nonzero) are evaluated by direct substitution.
• Standard algebraic limit: ${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$.
• Standard trigonometric limits: ${\lim }_{x\to 0}\frac{\sin x}{x}=1$, ${\lim }_{x\to 0}\frac{1-\cos x}{x}=0$, proved via the Sandwich Theorem.
• Limits of exponential/logarithmic functions (${\lim }_{x\to 0}(e^x-1)/x=1$, ${\lim }_{x\to 0}\log (1+x)/x=1$) are syllabus-named but not in this book; their derivatives are out of scope for this chapter entirely.
• Derivative: $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$, geometrically the slope of the tangent to $y=f(x)$.
• $(u\pm v{)}^{\prime }=u^{\prime }\pm v^{\prime }$; product rule $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$; quotient rule $(u/v{)}^{\prime }=(u^{\prime }v-u{v}^{\prime })/v^2$.
• $\frac{d}{dx}(x^n)=n{x}^{n-1}$; the same first-principles argument gives $\frac{d}{dx}(ax+b{)}^n=an(ax+b{)}^{n-1}$ without needing the general (formative-only) Chain Rule.
• $\frac{d}{dx}(\sin x)=\cos x$, $\frac{d}{dx}(\cos x)=-\sin x$ — the only two trig derivatives the book derives directly; $\tan x,\cot x,\sec x,\csc x$ follow from these via the quotient rule.