Limits and Derivatives revision notes

Calculus is the branch of mathematics that mainly deals with the study of change in the value of a function as the points in the domain change. The concept of derivatives stems from the need to find the instantaneous rate of change, such as finding the exact velocity of a falling body at a specific second rather than an average velocity over an interval.

The instantaneous velocity $v(t)$ at time $t=a$ is equal to the slope of the tangent to the distance-time curve at $t=a$. To precisely define this slope, the mathematical concept of a "limit" is required.

The limit of a function $f(x)$ as $x$ approaches $a$ is the expected value of $f(x)$ based on the values of $f(x)$ for points near $a$.

• Left Hand Limit (LHL): The expected value of $f(x)$ at $x=a$ given the values of $f(x)$ near $x$ to the left of $a$ (i.e., $x < a$). It is denoted as $\lim_{x \to a^-} f(x)$.
• Right Hand Limit (RHL): The expected value of $f(x)$ at $x=a$ given the values of $f(x)$ near $x$ to the right of $a$ (i.e., $x > a$). It is denoted as $\lim_{x \to a^+} f(x)$.
• Existence of Limit: If the right and left hand limits coincide, their common value is called the limit of $f(x)$ at $x = a$ and is denoted by $\lim_{x \to a} f(x)$. If they are different, the limit does not exist, even if the function is defined at that point. JEE Tip Always check LHL and RHL separately for piecewise functions, modulus functions, and greatest integer functions.

Let $f$ and $g$ be two functions such that both $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist.

1. Sum Rule: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$.
2. Difference Rule: $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$.
3. Product Rule: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$.
4. Quotient Rule: $\lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$.
5. Scalar Multiple Rule: $\lim_{x \to a} [\lambda \cdot f(x)] = \lambda \cdot \lim_{x \to a} f(x)$ for any real number $\lambda$.

A polynomial function is of the form $f(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$.

• The limit of a polynomial function at $x = a$ is simply the value of the function at $a$: $\lim_{x \to a} f(x) = f(a)$.

Standard Limit Formula for Powers

For any positive integer $n$ (and by extension, any rational number), $\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$. JEE Tip This is highly useful for fractional powers where binomial expansion is tedious.

Limits of trigonometric functions rely on the order properties of these functions:

• Theorem: If $f(x) \leq g(x)$ for all $x$ in the domain, and their limits exist at $a$, then $\lim_{x \to a} f(x) \leq \lim_{x \to a} g(x)$.
• Sandwich Theorem (Squeeze Theorem): Let $f, g$, and $h$ be real functions such that $f(x) \leq g(x) \leq h(x)$ for all $x$ in the common domain. If $\lim_{x \to a} f(x) = l = \lim_{x \to a} h(x)$, then $\lim_{x \to a} g(x) = l$. JEE Tip The Sandwich Theorem is the primary tool for solving limits involving oscillating functions like $\sin(1/x)$ combined with polynomials, e.g., $\lim_{x \to 0} x^2 \sin(1/x) = 0$.

Fundamental Trigonometric Inequality

For $0 < x < \frac{\pi}{2}$ (where $x$ is in radians), $\sin x < x < \tan x$.

Standard Trigonometric Limits

1. $\lim_{x \to 0} \frac{\sin x}{x} = 1$.
2. $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$.
3. $\lim_{x \to 0} \frac{\tan x}{x} = 1$. JEE Tip These limits are valid ONLY when $x$ is measured in RADIANS. If $x$ is in degrees, $\lim_{x \to 0} \frac{\sin x^\circ}{x} = \frac{\pi}{180}$.

The derivative of a function quantifies the rate of change of $f(x)$ with respect to $x$. Geometrically, the derivative of $f(x)$ at $x=a$ is the slope of the tangent to the curve $y=f(x)$ at the point $(a, f(a))$.

Derivative from First Principle

The derivative of $f$ at $x$ is defined as: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. It is denoted by $f'(x)$, $\frac{d}{dx}[f(x)]$, $\frac{dy}{dx}$, or $D(f(x))$.

Let $f(x) = u$ and $g(x) = v$ be functions whose derivatives exist.

1. Sum Rule: $(u + v)' = u' + v'$.
2. Difference Rule: $(u - v)' = u' - v'$.
3. Product Rule (Leibnitz Rule): $(uv)' = u'v + uv'$.
4. Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ (where $v \neq 0$). JEE Tip Trap alert! The negative sign in the numerator is specifically in front of the term $uv'$, which differentiates the denominator. Order matters!

1. Constant Function: $\frac{d}{dx}(a) = 0$.
2. Identity Function: $\frac{d}{dx}(x) = 1$.
3. Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ for any real number $n$.
4. Polynomial Function: $\frac{d}{dx}(a_n x^n + a_{n-1} x^{n-1} + \dots + a_0) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \dots + a_1$.
5. Trigonometric Functions:
   • $\frac{d}{dx}(\sin x) = \cos x$.
   • $\frac{d}{dx}(\cos x) = -\sin x$.
   • $\frac{d}{dx}(\tan x) = \sec^2 x$.
   • $\frac{d}{dx}(\cot x) = -\text{cosec}^2 x$.
   • $\frac{d}{dx}(\sec x) = \sec x \tan x$ [Derived using quotient rule on $1/\cos x$].
   • $\frac{d}{dx}(\text{cosec} x) = -\text{cosec} x \cot x$.

• L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ results in $0/0$ or $\infty/\infty$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the latter limit exists. JEE Tip Always check if the form is actually $0/0$ or $\infty/\infty$ before applying L'Hôpital's Rule. Applying it to finite forms gives incorrect answers!
• $1^\infty$ Form Limits: If $\lim_{x \to a} f(x) = 1$ and $\lim_{x \to a} g(x) = \infty$, then $\lim_{x \to a} f(x)^{g(x)} = e^{\lim_{x \to a} g(x)[f(x) - 1]}$.

• Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

• Average Velocity: $v_{avg} = \frac{\Delta s}{\Delta t} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$. Unit: m/s (meters per second).
• Instantaneous Rate of Change / Limit definition of Derivative: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.
• Standard Algebraic Limit: $\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$.
• Half-Angle identity limit evaluation: $1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right)$ is frequently used to prove $\lim_{x \to 0} \frac{1-\cos x}{x} = 0$.

• Limit Existence Constraint: $\lim_{x \to a} f(x)$ ONLY exists if the Left Hand Limit equals the Right Hand Limit.
• Quotient Rule Constraint: $\frac{d}{dx}\left(\frac{u}{v}\right)$ and limits of quotients $\lim \frac{f(x)}{g(x)}$ are valid strictly where the denominator $v \neq 0$ or $\lim g(x) \neq 0$ respectively.
• Rational Limit Division Constraint: When canceling factors like $(x-a)$ out of the form $0/0$, the mathematical assumption validating the cancellation is $x \to a$ implies $x \neq a$ (hence you are not dividing by exactly zero).

• Misconception: $f(a)$ is always equal to $\lim_{x \to a} f(x)$. Fact: The limit depends strictly on the values around $x=a$, not the value at $x=a$. The value $f(a)$ and the limit can be completely different, or one may exist while the other does not.
• Misconception: Infinite limit means limit exists. Fact: If $\lim_{x \to a} f(x) = \infty$, the limit technically does not exist because infinity is not a real number.
• Misconception: Radians and degrees are interchangeable in limits. Fact: The formulas $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and derivative $\frac{d}{dx}(\sin x) = \cos x$ STRICTLY assume $x$ is measured in radians. If $x$ is in degrees, the limit is $\pi/180$.
• Sign Convention in Quotient Rule: Always remember it is $(\text{Numerator}' \times \text{Denominator}) - (\text{Numerator} \times \text{Denominator}')$, divided by $\text{Denominator}^2$. Reversing the negative sign is a fatal algebraic error.

• $1^\infty$ limits: Used in virtually every JEE paper.
• Limits with Greatest Integer Functions $[x]$ and Fractional Parts $\{x\}$: These require breaking limits carefully into LHL and RHL because these functions jump at integer points.
• Evaluating Limits using Taylor/Maclaurin Series Expansion: For complex polynomial/trigonometric mixed fractions where L'Hôpital's rule becomes too lengthy.
• Derivatives of Implicit Functions & Inverse Trigonometric Functions.
• Checking continuity and differentiability via First Principles.

1. Derivation of $\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$ (for integer $n$)

• Method 1 (Factorization): Divide $(x^n - a^n)$ by $(x - a)$. $x^n - a^n = (x - a)(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + \dots + x a^{n-2} + a^{n-1})$. Taking the limit as $x \to a$ of the second bracket: $\lim_{x \to a} (x^{n-1} + a x^{n-2} + \dots + a^{n-1}) = a^{n-1} + a(a^{n-2}) + \dots + a^{n-1}$. There are $n$ terms, so the sum is $n a^{n-1}$.
• Method 2 (Binomial Theorem for $x \to a$ or $h \to 0$): Let $x = a + h$. Then $\lim_{h \to 0} \frac{(a+h)^n - a^n}{h}$. Expand $(a+h)^n$ using binomial theorem, the $a^n$ cancels, divide by $h$, and substituting $h=0$ leaves only $n a^{n-1}$.

2. Geometric Proof of $\sin x < x < \tan x$

• Consider a unit circle with center $O$ and an angle $x$ (in radians) such that $0 < x < \pi/2$.
• Area of $\Delta OAC < \text{Area of Sector } OAC < \text{Area of } \Delta OAB$.
• Area $\Delta OAC = \frac{1}{2} OA \cdot CD = \frac{1}{2} (1) \sin x = \frac{1}{2} \sin x$.
• Area of Sector $OAC = \frac{1}{2} r^2 x = \frac{1}{2} (1)^2 x = \frac{1}{2} x$.
• Area of $\Delta OAB = \frac{1}{2} OA \cdot AB = \frac{1}{2} (1) \tan x = \frac{1}{2} \tan x$.
• Therefore, $\sin x < x < \tan x$. Dividing by $\sin x$ gives $1 < \frac{x}{\sin x} < \frac{1}{\cos x}$, taking reciprocals gives $\cos x < \frac{\sin x}{x} < 1$, which proves $\lim_{x \to 0} \frac{\sin x}{x} = 1$ via Sandwich Theorem.

3. Derivative of $\sin x$ from First Principle

$f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$. Using the formula $\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$: $f'(x) = \lim_{h \to 0} \frac{2 \cos(x + \frac{h}{2}) \sin(\frac{h}{2})}{h}$. $= \lim_{h \to 0} \cos(x + \frac{h}{2}) \cdot \lim_{h \to 0} \frac{\sin(h/2)}{h/2} = \cos(x) \cdot 1 = \cos x$.