Quick Revision Notes: Limits and Derivatives

Class 11 Mathematics (MP Board)

Part 1: Limits (सीमाएँ)

1. Introduction to Limit

The limit of a function $f(x)$ as $x$ approaches $a$ is the value that the function approaches as $x$ gets closer and closer to $a$.

Notation: $\lim_{x \to a} f(x) = L$
Existence of Limit: A limit exists if and only if the Left Hand Limit (LHL) equals the Right Hand Limit (RHL). $$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = \lim_{x \to a} f(x)$$

2. Standard Limits (महत्वपूर्ण सीमाएँ)

$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n \cdot a^{n-1}$
$\lim_{x \to 0} \frac{\sin x}{x} = 1$ (where $x$ is in radians)
$\lim_{x \to 0} \frac{\tan x}{x} = 1$
$\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$
$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
$\lim_{x \to 0} \frac{a^x - 1}{x} = \log_e a$
$\lim_{x \to 0} \frac{\log(1 + x)}{x} = 1$

3. Algebra of Limits

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:

Sum Rule: $\lim_{x \to a} [f(x) \pm g(x)] = L \pm M$
Product Rule: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$
Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$, provided $M \neq 0$
Scalar Multiple: $\lim_{x \to a} [c \cdot f(x)] = c \cdot L$

Part 2: Derivatives (अवकलज)

1. Definition of Derivative (First Principle)

The derivative of a function $f(x)$ with respect to $x$ is denoted by $f'(x)$ or $\frac{dy}{dx}$ and is defined as: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

2. Algebra of Derivatives (बीजगणितीय नियम)

Let $f$ and $g$ be two functions.

Sum/Difference Rule: $\frac{d}{dx} [f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)]$
Product Rule (गुणन नियम): $\frac{d}{dx} [f(x) \cdot g(x)] = f(x) \cdot \frac{d}{dx}[g(x)] + g(x) \cdot \frac{d}{dx}[f(x)]$
Quotient Rule (भाग नियम): $\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{g(x) \cdot \frac{d}{dx}[f(x)] - f(x) \cdot \frac{d}{dx}[g(x)]}{[g(x)]^2}$, provided $g(x) \neq 0$

3. Standard Derivatives (मानक अवकलज)

Function $f(x)$	Derivative $\frac{d}{dx}[f(x)]$
$c$ (Constant)	$0$
$x^n$	$n \cdot x^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x \cdot \log_e a$
$\log_e x$	$\frac{1}{x}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \cdot \tan x$
$\csc x$	$-\csc x \cdot \cot x$

Quick Tips for MP Board Exam

Always check for indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying standard limit formulas. Factorization and rationalization are key techniques to resolve $0/0$ forms.
Remember to write the First Principle formula clearly when a question specifically asks to "find the derivative using the first principle".
Pay close attention to positive and negative signs while differentiating trigonometric functions (e.g., derivative of $\cos x$ is $-\sin x$).

Limits and Derivatives

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