Class 11 Maths Chapter 12 Limits and Derivatives – Revision Notes

A derivative is defined as the limit of a function’s average rate of change as the interval shrinks to zero. Chapter 12 of Class 11 Maths builds this idea from standard limits like sin x over x up to the derivative rules for powers, sine, and cosine.

Last Updated: September 23, 2026

Limits

• lim_x→a f(x): value f(x) approaches as x→a.
• Algebra of limits: sum/product/quotient rules apply term-wise.

Standard Limits

• lim_x→0 sinx/x = 1.
• lim_x→a (xⁿ−aⁿ)/(x−a) = naⁿ⁻¹.

Derivatives

• f'(x) = lim_h→0 [f(x+h)−f(x)]/h.
• d/dx(xⁿ) = nxⁿ⁻¹; d/dx(sinx)=cosx; d/dx(cosx)=−sinx; d/dx(constant)=0.

One-Line Summary

Limits describe the value a function approaches, forming the foundation for derivatives, which measure a function’s exact instantaneous rate of change (slope of the tangent) using a limiting process.

Frequently Asked Questions

What is the intuitive meaning of a limit?
A limit describes the value that a function approaches as the input gets closer and closer to a particular point, even if the function is not actually defined at that exact point.

How is the derivative of a function related to the concept of a limit?
The derivative at a point is defined as the limit of the average rate of change of the function as the interval shrinks to zero, which geometrically represents the slope of the tangent to the curve at that point.

Written by Satish

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