🔹 1. Introduction

Limits and Derivatives form the foundation of calculus. This chapter introduces how functions behave as inputs approach certain values and how to measure the rate of change.

👉 Used in:

• Physics (motion, velocity)
• Engineering
• Advanced calculus (Class 12 & JEE)

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🔹 2. Limits

✔️ Concept of Limit

Let f(x) be a function. The limit of f(x) as x approaches a is written as:

$\lim_{x \to a} f(x)$limx→a​f(x)

👉 It represents the value f(x) approaches as x gets close to a.

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✔️ Left-Hand and Right-Hand Limits

• Left-hand limit (LHL):

$\lim_{x \to a^-} f(x)$limx→a−​f(x)

• Right-hand limit (RHL):

$\lim_{x \to a^+} f(x)$limx→a+​f(x)

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✔️ Condition for Existence of Limit

Limit exists if:

👉 LHL = RHL

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🔹 3. Standard Limits

These are very important for CBSE & JEE:

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✔️ 1

$\lim_{x \to 0} \frac{\sin x}{x} = 1$limx→0​xsinx​=1

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✔️ 2

$\lim_{x \to 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}$limx→0​x21−cosx​=21​

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✔️ 3

$\lim_{x \to 0} \frac{\tan x}{x} = 1$limx→0​xtanx​=1

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✔️ 4

$\lim_{x \to 0} \frac{e^x – 1}{x} = 1$limx→0​xex−1​=1

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✔️ 5

$\lim_{x \to 0} \frac{a^x – 1}{x} = \ln a$limx→0​xax−1​=lna

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🔹 4. Algebra of Limits

If limits exist:

• Sum:

$\lim (f + g) = \lim f + \lim g$lim(f+g)=limf+limg

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• Product:

$\lim (fg) = (\lim f)(\lim g)$lim(fg)=(limf)(limg)

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• Quotient:

$\lim \frac{f}{g} = \frac{\lim f}{\lim g}$limgf​=limglimf​

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🔹 5. Important Techniques

✔️ Factorization
✔️ Rationalization
✔️ Using identities
✔️ Substitution

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🔹 6. Derivatives

✔️ Definition

Derivative represents rate of change of a function.

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✔️ Definition using Limit

$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$f′(x)=limh→0​hf(x+h)−f(x)​

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✔️ Notations

• f′(x)
• dy/dx
• Df(x)

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🔹 7. Derivatives of Basic Functions

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✔️ Constant

$\frac{d}{dx}(c) = 0$dxd​(c)=0

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✔️ Power Function

$\frac{d}{dx}(x^n) = nx^{n-1}$dxd​(xn)=nxn−1

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✔️ Trigonometric Functions

$\frac{d}{dx}(\sin x) = \cos x$dxd​(sinx)=cosx

$\frac{d}{dx}(\cos x) = -\sin x$dxd​(cosx)=−sinx

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✔️ Exponential Function

$\frac{d}{dx}(e^x) = e^x$dxd​(ex)=ex

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✔️ Logarithmic Function

$\frac{d}{dx}(\ln x) = \frac{1}{x}$dxd​(lnx)=x1​

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🔹 8. Rules of Differentiation

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✔️ Sum Rule

$(f + g)’ = f’ + g’$(f+g)′=f′+g′

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✔️ Product Rule

$(fg)’ = f’g + fg’$(fg)′=f′g+fg′

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✔️ Quotient Rule

$\left(\frac{f}{g}\right)’ = \frac{f’g – fg’}{g^2}$(gf​)′=g2f′g−fg′​

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🔹 9. Derivative of Composite Functions (Chain Rule – Basic Idea)

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$dxd​[f(g(x))]=f′(g(x))⋅g′(x)

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🔹 10. Applications of Derivatives

✔️ Rate of change
✔️ Tangent and normal
✔️ Velocity and acceleration
✔️ Optimization (advanced topics)

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🔹 11. Important Exam Points

✔️ Standard limits are very important
✔️ Definition-based derivative questions common
✔️ Practice algebraic simplifications
✔️ Trigonometric limits frequently asked
✔️ Derivative formulas must be memorized

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🔹 12. Common Mistakes

❌ Direct substitution in limits without simplification
❌ Forgetting LHL ≠ RHL condition
❌ Sign mistakes in derivatives
❌ Confusing product and chain rule

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🔹 13. Practice Questions

1. Evaluate limits using identities
2. Prove standard limits
3. Find derivative using definition
4. Differentiate composite functions
5. Solve application-based problems

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🔹 14. Quick Revision Sheet

• lim x→0 (sinx/x) = 1
• f′(x) = lim h→0 [f(x+h) − f(x)]/h
• d/dx (xⁿ) = nxⁿ⁻¹
• d/dx (sinx) = cosx
• d/dx (eˣ) = eˣ
• Product rule: f’g + fg’

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🔹 15. Conclusion

Limits and Derivatives is the foundation of calculus. Strong understanding here will make Class 12 topics like:

• Continuity
• Differentiability
• Integration

👉 much easier and scoring.