Limits and Derivatives Class 11 CBSE Maths Chapter 13

Class 11 CBSE Maths Syllabus
Chapter 1:Sets
Chapter 2: Relations and Functions
Chapter 3: Trigonometric Functions
Chapter 4: Principles of Mathematical Induction
Chapter 5: Complex Numbers and Quadratic Equations
Chapter 6: Linear Inequalities
Chapter 7: Permutations and Combinations
Chapter 8: Binomial Theorem
Chapter 9: Sequences and Series
Chapter 10: Straight lines
Chapter 11: Conic Sections
Chapter 12: Introduction to 3D
Chapter 13: Limits and Derivatives
Chapter 14: Mathematical Reasoning
Chapter 15: Stastics
Chapter 16: Probability

Introduction

What are the limits?

In mathematics, a limit is a value that approaches some value.

Consider the function f(x) = x2

Here, x takes values very close to 0, so the value of f(x) also moves towards zero.

We can say that, x0f(x)=0

In general:

As xa, f(x)l, then then l is called the limit of the function f(x) which is symbolically written as xaf(x)=l

For all the limits, the value at which function should assume at a given point x=a did not really depend on how is x tending to a. There are two ways x could approach a number a, either from the left or from the right, i.e., all the values of x near a could be less than a or could be greater than a.

The two types of limits – the right-hand limit and the left-hand limit describe how x approaches a.

The value of the right-hand limit of a function f(x) is decided by the values of f(x) when x tends to a from right.
The value of the left-hand limit of a function f(x) is decided by the values of f(x) when x tends to a from left.

If the right and the left-hand limits coincide, we call that common value the limit of f(x) at x = a and denote it by xaf(x)

What are Derivatives?

A derivative is a way to show the rate of change, i.e., the rate by which a function is changing at a given point. For example,

People maintaining a reservoir need to know when the reservoir will overflow knowing the depth of the water at several instances of time.
Rocket scientists need to compute the precise velocity with which the satellite needs to be shot out from the rocket knowing the height of the rocket at various times.
Financial institutions need to predict the changes in the value of a particular stock knowing its present value.

Limits

This section covers left-hand and right-hand limits, as well as solved problems on limits. When an ice cube is thrown into a glass of hot water and the temperature is recorded over time, it is an illustration of a limit as time approaches infinity. The temperature steadily reaches the room temperature where the glass was stored during the time period monitored.

Limits of Trigonometric Functions

This section covers function theorems and prepositions, which are useful in determining the limits of several trigonometric functions.

The miscellaneous exercise of Chapter 13 covers the questions from all the important topics included in the chapter. Below we have summarised a few points to remember to help students prepare for this chapter.