Limits — Complete Formula

Basic Definition of Limit

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Limit	[\lim_{x \to a} f(x) = L]	[f(x)] = function, [a] = point, [L] = limiting value	Value approached by function ⭐

💡 Memory Hint:
Limit = approaching value, not necessarily equal

Concept	Formula	Symbols Meaning	Key Notes
LHL	[\lim_{x \to a^-} f(x)]	Approach from left	Important ⭐
RHL	[\lim_{x \to a^+} f(x)]	Approach from right	Important ⭐
Existence	[\text{LHL} = \text{RHL}]	—	Limit exists ⭐

Concept	Formula	Symbols Meaning	Key Notes
Trigonometric	[\lim_{x \to 0} \dfrac{\sin x}{x} = 1]	—	Most important ⭐
Cosine	[\lim_{x \to 0} \dfrac{1 – \cos x}{x^2} = \dfrac{1}{2}]	—	Frequently used ⭐
Tangent	[\lim_{x \to 0} \dfrac{\tan x}{x} = 1]	—	Derived from sin/cos
Exponential	[\lim_{x \to 0} \dfrac{e^x – 1}{x} = 1]	—	Important ⭐
Logarithmic	[\lim_{x \to 0} \dfrac{\ln(1 + x)}{x} = 1]	—	Standard ⭐
General Form	[\lim_{x \to 0} \dfrac{a^x – 1}{x} = \ln a]	[a] = constant	Advanced ⭐

💡 Memory Hint:
All standard limits → reduce to base forms

Concept	Formula	Symbols Meaning	Key Notes
Polynomial	Substitute directly	—	Works if no indeterminate form ⭐
Rational Function	Simplify first	—	Factorization important

Form	Method	Symbols Meaning	Key Notes
[\dfrac{0}{0}]	Factorization / expansion	—	Most common ⭐
[\dfrac{\infty}{\infty}]	Divide by highest power	—	Important ⭐
[0 \cdot \infty]	Convert to fraction	—	Trick-based
[\infty – \infty]	Rationalization	—	Careful simplification ⚠️
[0^0, 1^\infty, \infty^0]	Log method	—	Advanced ⭐

💡 Memory Hint:
Indeterminate → transform to solvable form

Concept	Formula	Symbols Meaning	Key Notes
Sum	[\lim(f + g) = \lim f + \lim g]	[f, g] = functions	Basic law ⭐
Product	[\lim(fg) = (\lim f)(\lim g)]	—	Always holds
Quotient	[\lim\dfrac{f}{g} = \dfrac{\lim f}{\lim g}]	[\lim g \ne 0]	Important ⚠️

Concept	Method	Symbols Meaning	Key Notes
Algebraic Simplification	Factor numerator/denominator	—	Used in [0/0] ⭐

Concept	Method	Symbols Meaning	Key Notes
Conjugate Method	Multiply by conjugate	—	Useful for roots ⭐

Concept	Method	Symbols Meaning	Key Notes
Direct Substitution	Replace value if valid	—	Only when no indeterminate form ⭐

Concept	Formula	Symbols Meaning	Key Notes
[e^x]	[e^x = 1 + x + \dfrac{x^2}{2!} + …]	—	Useful ⭐
[\sin x]	[x – \dfrac{x^3}{3!} + …]	—	Approximation ⭐
[\cos x]	[1 – \dfrac{x^2}{2!} + …]	—	Important ⭐

Concept	Formula	Symbols Meaning	Key Notes
Polynomial	Highest power dominates	—	Key idea ⭐
Exponential vs Polynomial	Exponential grows faster	—	Important ⭐
Standard Result	[\lim_{x \to \infty} \left(1 + \dfrac{1}{x}\right)^x = e]	—	Very important ⭐

Concept	Formula	Symbols Meaning	Key Notes
Theorem	If [g(x) \le f(x) \le h(x)] and limits of [g, h] same → limit of [f] same	—	Useful in tricky limits ⭐

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