Limits and Derivatives

Limits and derivatives are extremely crucial concepts in maths whose application is not only limited to maths but are also present in other subjects like physics. In this article, the complete concepts of limits and derivatives along with their properties, and formulas are discussed. Below is an overview of all the concepts that are covered in the limits and derivatives class 11.

  • Limits and Derivatives Introduction
  • About the Limit of a Function
    • Limit Formula
    • Properties of Limits
    • Example Question From Limits
  • About the Derivative of a Function
    • Derivatives Formula
    • Properties of Derivatives
    • Example Question From Derivatives
  • Practice Questions From Limits and Derivatives

Limits and Derivatives Introduction

Sir Issac Newton, based on his concepts of rate and change, set forth the basic laws of differential calculus and integral calculus came forth as the reverse process. The basics of differentiation and calculus serve as the foundation for advanced mathematics, modern physics and various other branches of modern sciences and engineering. Limits and derivatives class 11 serve as the entry point to calculus for CBSE students.

Limits of a Function

In Mathematics, a limit is defined as a value that a function approaches as the input approaches some value. Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity.

Limits Formula:

To express the limit of a function, we represent it as:

\(\displaystyle {\lim _{n\to c}f(n)=L}\)

Properties of Limits:

Let p and q be two functions and a be a value such that \(\displaystyle{\lim_{x \to a}p(x)}\) and \(\displaystyle{\lim_{x \to a}q(x)}\) exists.

1.\(\displaystyle{\lim_{x \to a}[p(x) + g (x)] = \lim_{x \to a}p(x) + \lim_{x \to a}g (x)}\)

2.\(\displaystyle{\lim_{x \to a}[p(x) – g (x)] = \lim_{x \to a}p(x) – \lim_{x \to a}g (x)}\)

3.For every real number k,

\(\displaystyle{\lim_{x \to a}[k p(x)] = k \lim_{x \to a}p(x)}\)

4.\(\displaystyle{\lim_{x \to a}[p(x)\; q(x)] = \lim_{x \to a}p(x) \times \lim_{x \to a}q(x)}\)

5.\(\displaystyle{\lim_{x \to a}\frac{p(x)}{q(x)}} =\frac{\displaystyle{\lim_{x \to a}p(x)}}{\displaystyle{\lim_{x \to a}q(x)}}\)

Also, check out the limit formula page here to learn more about it in detail. The formula is explained using solved examples which can help to understand the problem-solving startegies.

Example Question From Limits Concept

Example: Find \(\lim_{x\rightarrow3}x+3\)

Solution:

\(\lim_{x\rightarrow3}x+3=3+3=6\)

Derivatives of a Function

A derivative refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount. The derivative of a function is represented as mentioned in the below-given formula.

Derivative Formula

\(\displaystyle{\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}}\)

For the function f, its derivative is said to be f'(x) given the equation above exists. Check out all the derivative formulas here related to trigonometric functions, inverse functions, hyperbolic functions, etc.

Properties of Derivatives

Since the very definition of derivatives involves limits in a rather direct fashion, we expect the rules of derivatives to follow closely that of limits as given below:

1.\(\frac{d}{dx}\;[p(x)+q(x)]= \frac{d}{dx}\;(p(x))+ \frac{d}{dx}\;(q(x))\)

2.\(\frac{d}{dx}\;[p(x)-q(x)]= \frac{d}{dx}\;(p(x))- \frac{d}{dx}\;(q(x))\)

3.\(\frac{d}{dx}\;[p(x)\times q(x)]= \frac{d}{dx}\;[p(x)]\;q(x) + p(x)\;\frac{d}{dx}\;[q(x)]\)

4.\(\frac{d}{dx}\left [\frac{p(x)}{q(x)} \right ]= \frac{\frac{d}{dx}\;[p(x)]\;q(x) – p(x)\;\frac{d}{dx}\;[q(x)] }{(g(x))^2}\)

Example Question From Derivatives Concept

Example: What is \(\frac{d}{dx}(x^{3})\)?

Solution:

Since the function is a constant, \(\frac{d}{dx}(x^{3})=0\)

Practice Questions From Limits and Derivatives

  1. Evaluate \(\lim_{x\rightarrow3}\frac{x^{2}-9}{x-3}\)
  2. \(\lim_{x\rightarrow1}\frac{x^{2}+2x+5}{x^{2}+1}\)
  3. \(\lim_{x\rightarrow0}\frac{sin(a+x)-sin(a-x)}{x}\)
  4. Calculate \(\frac{dy}{dx}\;at\;x=1\;if\;y=\sqrt{x}+\frac{1}{\sqrt{x}}\)
  5. Find the derivative of y = 1 – x2 + x – 3x4

Also, read

Keep visiting BYJU’S to get more such maths lessons along with notes, preparation tips, and other study materials like syllabus, sample papers, and revision notes.

Practise This Question

When 3x3+2x2x+4 is divided by 3+2x the remainder is 18.

Leave a Comment

Your email address will not be published. Required fields are marked *