Limit definition $\lim_{x \to a} f(x) = L$ if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$ L'Hôpital's Rule $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ If form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ Standard Limit: $\frac{\sin x}{x}$ $\lim_{x \to 0} \frac{\sin x}{x} = 1$ Standard Limit: $\frac{\tan x}{x}$ $\lim_{x \to 0} \frac{\tan x}{x} = 1$ Standard Limit: $\frac{e^x-1}{x}$ $\lim_{x \to 0} \frac{e^x-1}{x} = 1$ Standard Limit: $\frac{\ln(1+x)}{x}$ $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$ Standard Limit: $(1+x)^{1/x}$ $\lim_{x \to 0} (1+x)^{1/x} = e$ Standard Limit: $(1/x)^x$ $\lim_{x \to \infty} (1+\frac{1}{x})^x = e$ Continuity at $x=a$ $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$ Function must be defined at $a$ Differentiability at $x=a$ LHD = RHD, i.e., $\lim_{h \to 0^-} \frac{f(a+h)-f(a)}{h} = \lim_{h \to 0^+} \frac{f(a+h)-f(a)}{h}$ Function must be continuous at $a$

Short Tricks & Substitution Hacks

For limits of the form $1^\infty$, use $\lim_{x \to a} [f(x)]^{g(x)} = e^{\lim_{x \to a} g(x)[f(x)-1]}$.
For functions involving $|x|$, check continuity/differentiability by evaluating LHL, RHL, LHD, RHD separately.
When L'Hôpital's rule is applicable, differentiate numerator and denominator until the indeterminate form is resolved.

PYQ Trend Analysis (2021–2024)

Top 3 Subtopics: L'Hôpital's Rule, Continuity of piecewise functions, Differentiability of functions involving modulus.
Time-killer: Limits that require multiple applications of L'Hôpital's rule or algebraic manipulation before applying standard formulas.

Must-Do Question Profiles

Q1: Evaluate $\lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1-\cos 2x)^2}$.

Q2: For what values of $a$ and $b$ is the function $f(x)$ continuous and differentiable at $x=1$? $f(x) = \begin{cases} x^2+3x+a, & x \le 1 \\ bx+2, & x > 1 \end{cases}$

Q3: If $f(x) = \begin{cases} x^2 \sin(1/x), & x \ne 0 \\ 0, & x = 0 \end{cases}$, examine the differentiability of $f(x)$ at $x=0$.

Common Trap Alerts

Assuming differentiability implies continuity (it's the other way around).
Incorrectly applying L'Hôpital's Rule when the limit form is not $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
Errors in evaluating limits involving trigonometric functions, especially when $x \to 0$.

8. Differentiation and Applications

Golden Formula Table

Concept	Formula/Definition	Conditions
Power Rule	$\frac{d}{dx}(x^n) = nx^{n-1}$	$n \in R$
Product Rule	$\frac{d}{dx}(uv) = u'v + uv'$