What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

What does \(\lim_{x\to a}f(x)=l\) mean?

As \(x\) takes values closer and closer to \(a\) (from either side), \(f(x)\) gets arbitrarily close to \(l\).

When does a limit at \(x=a\) exist?

The limit exists iff the left-hand limit and the right-hand limit at \(a\) are equal.

How is the limit of a polynomial at \(x=a\) found?

By direct substitution: \(\lim_{x\to a}f(x)=f(a)\) for a polynomial function.

Define the derivative of \(f\) at \(x\) from first principles.

\(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\), when this limit exists.

State the product rule for derivatives.

\((fg)'(x)=f'(x)g(x)+f(x)g'(x)\).

State the quotient rule for derivatives.

\(\left(\dfrac{f}{g}\right)'(x)=\dfrac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}\), where \(g(x)\ne0\).

Give the derivative of \(x^n\).

\(\dfrac{d}{dx}(x^n)=nx^{n-1}\) for a positive integer \(n\).

What is the derivative of \(\sin x\) at \(x=0\)?

From first principles it equals \(\lim_{h\to0}\dfrac{\sin h}{h}=1\), i.e. \(\cos 0=1\).

Limits and Derivatives

Class 11 Mathematics · Chapter 12

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Every formula of Ch 1–14, with its condition

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CBSE Class 11 · MathematicsTrending

Every formula of Ch 1–14, with its condition

Marks booster + cheatsheet: key concepts in 2 lines each, common traps
25 marks: Unit II, Ch 4–8
23 marks: Unit I, Ch 1–3
Common traps, corrected
Conic table and standard values
Graphs plotted from the book's own tables

Page 1 free about 8 min read, Full kit about 55 min read

₹159See the full kit

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Summary

This chapter introduces calculus, the study of change. Starting from average velocities of a falling body, it builds an intuitive idea of the derivative as an instantaneous rate of change. The limit \(\lim_{x\to a}f(x)\) is the value \(f(x)\) approaches as \(x\) approaches \(a\); it exists only when the left-hand limit \(\lim_{x\to a^-}f(x)\) and right-hand limit \(\lim_{x\to a^+}f(x)\) coincide. Algebra of limits lets the limit of a sum, difference, product or quotient be computed from the limits of the parts; for polynomials and rational functions the limit is found by substitution (where the denominator is non-zero). Two standard trigonometric limits are \(\lim_{x\to0}\dfrac{\sin x}{x}=1\) and \(\lim_{x\to0}\dfrac{1-\cos x}{x}=0\). The derivative of \(f\) at \(x\) is defined from first principles as \(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\). The algebra of derivatives gives sum, difference, product and quotient rules, and yields standard results such as \(\dfrac{d}{dx}(x^n)=nx^{n-1}\), \(\dfrac{d}{dx}\sin x=\cos x\) and \(\dfrac{d}{dx}\cos x=-\sin x\).

Intuitive idea of the derivativeLimits: left- and right-hand limitsAlgebra of limitsLimits of polynomials and rational functionsLimits of trigonometric functionsDerivative from first principles and algebra of derivatives

Key terms

Limit

\(\lim_{x\to a}f(x)=l\) means \(f(x)\) approaches \(l\) as \(x\) approaches \(a\).

Left- and right-hand limits

The values approached as \(x\to a^-\) and \(x\to a^+\); the limit exists when they are equal.

Algebra of limits

Limit of a sum, difference, product or quotient equals the corresponding combination of the limits.

Standard trig limit

\(\lim_{x\to0}\dfrac{\sin x}{x}=1\) and \(\lim_{x\to0}\dfrac{1-\cos x}{x}=0\).

Derivative (first principle)

\(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\).

Standard derivatives

\(\dfrac{d}{dx}x^n=nx^{n-1}\), \(\dfrac{d}{dx}\sin x=\cos x\), \(\dfrac{d}{dx}\cos x=-\sin x\).

Extra questions & answers

Try one from this chapter

A limit \(\lim_{x\to a}f(x)\) exists when:

Flashcards & interactive practice