Limits and Derivatives
Class 11 Mathematics · Chapter 12
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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: