LIMITS AND DERIVATIVES

⭐ Limits and Derivatives — Chapter Overview

📘 Introduction

Limits and derivatives are the foundation of calculus. They help us understand how functions behave near certain points and how quantities change with respect to one another. This chapter introduces the basic ideas needed to study change, motion, and rate of growth.

LIMITS

A limit describes the value that a function approaches as the input gets closer and closer to a particular number.
It helps when direct substitution is not possible (like 0/0 form).

Example:

lim⁡(x→2) (3x+1)=7

As x gets closer to 2, the value of the expression approaches 7.

👉RIGHT HAND LIMIT (RHL)

The right-hand limit of a function at x=a is the value that the function gets closer to when x approaches to a from the right, meaning from numbers greater than a.

It is written as:

lim⁡ x→a+ f(x)

👉LEFT HAND LIMIT (LHL):

The Left-Hand Limit of a function at x=a is the value that the function approaches as x approaches a from the left side, that is, from values less than a.
It is written as:

lim ⁡x→a− f(x)

⭐VERTICAL ASYMPTOTE :

The line x=a is called vertical asymptote of the curve y=f(x) , if

lim x--->a f(x)=♾️.

TO FIND VERTICAL ASYMPTOTE -

Set the denominator =0
The values of x that make denominator zero ( and not cancelled ) give vertical asymptotes.

Example:

> f(x) = x^2/x^2 - 1

= x^2 - 1 = 0

= x^2 = 1

= x = +1 , x= -1

⭐ HORIZONTAL ASYMPTOTE :

The line y=L is called horizontal asymptote of the curve y=f(x) , if

lim x--->♾️ f(x)=L .

Example :

> f(x) = x^2/x^2 - 1

= lim x--->♾️ x^2 [1] / x^2 [1 - 1/x^2 ]

= lim x--->♾️ 1 / 1-0

= 1 = L = y .

VERTICAL ASYMPTOTE

HORIZONTAL ASYMPTOTE

✨CONTINUITY

A function f is said to be continuous at a number 'a' if :

limit exist at 'a'
f(a) is defined
lim x→a f(x)= f(a)

✨DISCONTINUITY

A function f is said to be discontinuous at a number 'a' if :

limit doesn't exist at 'a'
f(a) is undefined at 'a'
lim x→a f(x) ≠ f(a)

🔗DERIVATIVES

The derivative of a function f with respect to the variable x in the function f' whose value at x is

dy/dx = f'(x)=lim h→0 f(x+h) - f(x)/h proved limit exists .

⚫TANGENT LINE :

The line that touches the curve at only one point but not pass through it .

⚫SLOPE OF A TANGENT :

lim h--->0 f( a+h ) - f(a) / h .

⚫SECANT LINE :

The line which cuts the curve twice is called a secant line .

⚫SLOPE OF A SECANT

f( a+h ) - f(a) / h .

💭IMP NOTE

Every differentiable function is continuous function but every continuous function may not be differentiable

where function 'f' can't be differentiable

1.Corner or kink : It is not continuous because the tangent can't be drawn

2.Jump: It is discontinuous because LHL is not equal to RHL .

3.Vertical tangent: It is not continuous because lim x--->a f(a)=♾️.

🌠DIFFERENTIATION

The differentiation of a function is the process of finding its derivative, which measures the instantaneous rate at which the function's value changes with respect to a change in its independent variable.

Derivative

A derivative tells you how fast something is increasing or decreasing at a particular moment.