Limits and Derivatives — Class 11 Mathematics

Core Concepts

A limit describes the value a function approaches as x approaches a given point. A derivative measures instantaneous rate of change and geometrically represents the slope of a tangent.

Worked Examples

lim x→2 (x²−4)/(x−2)=lim x→2 (x+2)=4, after factoring x²−4=(x−2)(x+2).

If f(x)=x², then f′(x)=2x. At x=3, the derivative is 6, meaning the tangent slope there is 6.

Explained MCQs

1. d(x³)/dx=3x².
2. The derivative represents instantaneous rate of change.
3. For f(x)=5, f′(x)=0 because a constant has no change.

Practice

1. Evaluate lim x→3 (x²−9)/(x−3).
2. Differentiate 4x³−5x+7.
3. Find the slope of y=x²+2x at x=1.