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Revision Checklist Last-Minute JEE Prep

Limits Revision Checklist: Last Minute Guide for JEE

All formulas, techniques, and common results in one place for JEE Main & Advanced

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Basic Formulas Standard Limits Techniques Indeterminate Forms Special Limits JEE Tips Common Errors Practice

1. Basic Limit Formulas & Properties

Limit Definition

$\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$

$$ \forall \epsilon > 0, \exists \delta > 0 \text{ such that } 0 < |x-a| < \delta \Rightarrow |f(x)-L| < \epsilon $$

Basic Properties

Sum/Difference:
$\lim[f(x) \pm g(x)] = \lim f(x) \pm \lim g(x)$

Product:
$\lim[f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x)$

Quotient:
$\lim\left[\frac{f(x)}{g(x)}\right] = \frac{\lim f(x)}{\lim g(x)}$ if $\lim g(x) \neq 0$

Constant Multiple:
$\lim[c \cdot f(x)] = c \cdot \lim f(x)$

Important Theorems

Sandwich Theorem: If $g(x) \leq f(x) \leq h(x)$ near $a$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$

Limit Existence: $\lim_{x \to a} f(x)$ exists iff both left-hand and right-hand limits exist and are equal

2. Standard Limits (Must Memorize)

Trigonometric Limits

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$\lim_{x \to 0} \frac{\tan x}{x} = 1$

$\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$

Exponential & Logarithmic Limits

$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

$\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$

$\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$

$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

Important Algebraic Limits

$\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$

$\lim_{x \to \infty} \frac{1}{x^n} = 0$ (n > 0)

$\lim_{x \to 0} \frac{(1 + x)^n - 1}{x} = n$

$\lim_{x \to \infty} \frac{\sin x}{x} = 0$

3. Limit Evaluation Techniques

Direct Substitution

Always try direct substitution first. If it works, you're done!

Example: $\lim_{x \to 2} (3x^2 - 2x + 1) = 3(4) - 2(2) + 1 = 9$

Factorization

Use when you get 0/0 form. Factor and cancel common terms.

Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} (x+2) = 4$

Rationalization

Use for expressions with square roots in 0/0 form.

Example: $\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x} = \lim_{x \to 0} \frac{(\sqrt{1+x}-1)(\sqrt{1+x}+1)}{x(\sqrt{1+x}+1)} = \lim_{x \to 0} \frac{1}{\sqrt{1+x}+1} = \frac{1}{2}$

L'Hôpital's Rule

For 0/0 or ∞/∞ forms: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

⚠️ Remember: Only for 0/0 or ∞/∞ forms! Check conditions before applying.

4. Indeterminate Forms & Their Solutions

0/0 Form

Techniques:

Factorization
Rationalization
Standard limits
L'Hôpital's Rule
Series expansion

∞/∞ Form

Techniques:

Divide numerator and denominator by highest power
L'Hôpital's Rule
Comparison of growth rates

Example: $\lim_{x \to \infty} \frac{3x^2 + 2x}{2x^2 - 1} = \lim_{x \to \infty} \frac{3 + \frac{2}{x}}{2 - \frac{1}{x^2}} = \frac{3}{2}$

∞ - ∞ Form

Techniques:

Rationalization
Take LCM and combine
Factor out common terms

1^∞ Form

Standard Approach: Use $\lim_{x \to a} [f(x)]^{g(x)} = e^{\lim_{x \to a} [f(x)-1]g(x)}$

Example: $\lim_{x \to 0} (1 + x)^{1/x} = e^{\lim_{x \to 0} \frac{1}{x} \cdot x} = e^1 = e$

5. Special Limits for JEE Advanced

Limits with Parameters

$\lim_{x \to 0} \frac{\sin ax}{bx} = \frac{a}{b}$

$\lim_{x \to 0} \frac{\tan mx}{nx} = \frac{m}{n}$

$\lim_{x \to 0} \frac{e^{ax} - 1}{bx} = \frac{a}{b}$

Piecewise Limits

Always check left-hand and right-hand limits separately at break points.

Example: For $f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x & \text{if } x \geq 1 \end{cases}$
$\lim_{x \to 1^-} f(x) = 1$, $\lim_{x \to 1^+} f(x) = 2$ ∴ Limit doesn't exist at x=1

Limits at Infinity

$\lim_{x \to \infty} \frac{\text{Polynomial}}{\text{Polynomial}}$ = Compare degrees

$\lim_{x \to \infty} \frac{\ln x}{x^p} = 0$ (p > 0)

$\lim_{x \to \infty} \frac{e^x}{x^n} = \infty$

$\lim_{x \to \infty} \frac{a^x}{x^n} = \infty$ (a > 1)

6. JEE Problem Solving Tips

Systematic Approach

Always try direct substitution first
Identify the form (0/0, ∞/∞, etc.)
Choose appropriate technique based on form
Verify your answer makes sense
For piecewise functions, check both sides

Time-Saving Techniques

Memorize standard limits for quick recognition
Use symmetry when possible
For trigonometric limits, know small angle approximations
Practice mental calculation of simple limits

7. Common Errors to Avoid

Algebraic Mistakes

❌ Wrong: $\lim_{x \to 0} \frac{\sin x}{x} = 0$ (Cancelling incorrectly)
✅ Correct: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (Standard limit)

L'Hôpital Misuse

❌ Wrong: Applying L'Hôpital to forms other than 0/0 or ∞/∞
✅ Correct: Only use for 0/0 or ∞/∞ forms after verification

Piecewise Function Errors

❌ Wrong: Not checking both sides for piecewise functions
✅ Correct: Always compute $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ separately

📝 Quick Practice Test

Evaluate these limits to test your understanding:

1. $\lim_{x \to 0} \frac{\sin 3x}{\tan 2x}$

2. $\lim_{x \to 1} \frac{x^2 - 3x + 2}{x^2 - 1}$

3. $\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{3x}$

4. $\lim_{x \to 0} \frac{e^{2x} - 1}{\sin x}$

Answers: 1. $\frac{3}{2}$, 2. $-\frac{1}{2}$, 3. $e^6$, 4. $2$

Final Revision Notes

🎯 Key Concepts to Remember

Always try direct substitution first
Know your standard limits by heart
Identify the form before choosing technique
For piecewise functions, check both sides
Practice multiple techniques for each form

🚀 Exam Day Strategy

Solve limits in this order: Direct → Factor → Rationalize → L'Hôpital
Double-check algebraic manipulations
Verify L'Hôpital conditions before applying
If stuck, try graphical interpretation
Manage time - move on if taking too long

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