Summary & JEE Exam Tips

Chapter Summary: Key Formulas and Concepts

Limits

• Existence: A limit exists at $x=a$ if and only if the Left-Hand Limit (LHL) equals the Right-Hand Limit (RHL). $$\lim_{x \to a} f(x) \text{ exists} \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$$
• Indeterminate Forms: $\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \times \infty, 1^\infty, 0^0, \infty^0$.
• L'Hôpital's Rule: For $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms, $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$.

Standard Limits (as $x \to 0$)

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$, $\lim_{x \to 0} \frac{\tan x}{x} = 1$
• $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
• $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$
• $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$
• $\lim_{x \to 0} (1+x)^{1/x} = e$
• $\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}$
• $\lim_{x \to a} \frac{x^n - a^n}{x-a} = na^{n-1}$

Series Expansions (for $x$ near 0)

• $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
• $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$
• $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$
• $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$

Continuity

• Condition at $x=a$: A function is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$.
• This requires three things: $f(a)$ is defined, the limit exists, and they are equal.

Derivatives

• First Principle: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
• Differentiability: A function is differentiable at $x=a$ if the Left-Hand Derivative (LHD) equals the Right-Hand Derivative (RHD).
• Key Relation: Differentiability implies Continuity. (Continuity does NOT imply Differentiability).

Rules of Differentiation

• Product Rule: $(uv)' = u'v + uv'$
• Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$
• Chain Rule: If $y=f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

JEE Exam Tips & Strategies

1. Master Standard Limits: For speed and accuracy, recognizing and applying standard limits is almost always faster than using L'Hôpital's Rule. Use L'Hôpital's Rule as a reliable backup, not your primary tool.

2. Use Series Expansions for $x \to 0$: For complicated limits where $x \to 0$, especially those involving trigonometric, exponential, and logarithmic functions, using Maclaurin series expansions is a very powerful and quick method. Often, you only need the first few non-zero terms.

3. The $1^\infty$ Form Shortcut: For limits of the form $\lim_{x \to a} f(x)^{g(x)}$ which results in $1^\infty$, use the direct formula: $$L = e^{\lim_{x \to a} g(x)[f(x)-1]}$$ This is much faster than the logarithmic method.

4. Check LHD and RHD for Piecewise and Modulus Functions: When checking for differentiability of piecewise functions or functions involving absolute values (like $|x-a|$), always check the LHD and RHD at the boundary points. Don't just differentiate the formulas directly. A sharp corner exists if LHD $\neq$ RHD.

5. Simplify Before Differentiating: Before applying a complex rule like the quotient rule, see if the expression can be simplified algebraically first. For example, $\frac{x^2+x}{x}$ should be simplified to $x+1$ before differentiating.

6. Logarithmic Differentiation: For functions of the form $f(x)^{g(x)}$ or functions involving a complex product of terms, use logarithmic differentiation. Take the natural log of both sides, use log properties to simplify, and then differentiate implicitly.

7. Differentiability and Continuity Link: Remember that if a function is not continuous at a point, it cannot be differentiable there. This can be a quick way to eliminate options or solve problems. If asked to check for differentiability, first perform a quick mental check for continuity.