Limits and Derivatives revision notes

Introduction & Intuitive Idea of Limits

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.

Left Hand and Right Hand Limits

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.

Algebra of Limits

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 $\lambda$.

Limits of Polynomials and Rational Functions

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 $a$: $\lim_{x \to a} f(x) = f(a)$.
• A rational function is $f(x) = \frac{g(x)}{h(x)}$, where $g(x)$ and $h(x)$ are polynomials.
  • If $h(a) \neq 0$, then $\lim_{x \to a} f(x) = \frac{g(a)}{h(a)}$.
  • If $h(a) = 0$ and $g(a) \neq 0$, the limit does not exist.
  • If $h(a) = 0$ and $g(a) = 0$, this is a $0/0$ indeterminate form. To evaluate, factorize $g(x)$ and $h(x)$ as $g(x) = (x-a)^k g_1(x)$ and $h(x) = (x-a)^l h_1(x)$, cancel the common $(x-a)$ terms, and then evaluate. → [JEE TIP] We can safely cancel $(x-a)$ from numerator and denominator because the limit operation implies $x \to a$, which strictly means $x \neq a$.

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

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, $\lim_{x \to 0} \frac{\sin x^\circ}{x} = \frac{\pi}{180}$.

Derivatives & The First Principle

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 $D(f(x))$.

Algebra of Derivatives

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!

Standard Derivatives

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 $1/\cos x$].
   • $\frac{d}{dx}(\text{cosec} x) = -\text{cosec} x \cot x$.

JEE Advanced Extensions (Added Topics)

• 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 $\lim_{x \to a} f(x)^{g(x)} = e^{\lim_{x \to a} g(x)[f(x) - 1]}$.
• Standard Maclaurin Series Expansions: Extremely powerful for complex $0/0$ limits:
  • $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$
  • $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$
  • $\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \dots$
  • $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$
  • $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$
• Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

Key Concepts & Definitions

Calculus:

Mathematics explaining the course of nature, dealing with the study of change in a function's value as points in its domain change.

Derivative:

The instantaneous rate of change of a function, representing the slope of the tangent line to the function's curve at a specific point.

Limit:

The expected value lll that a function f(x)f(x)f(x) approaches as the independent variable xxx approaches a given value aaa.

First Principle:

The foundational definition of differentiation evaluating limits of the difference quotient as the interval approaches zero.

Conditions & Limitations

• 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).

Previous Year JEE Topics

• $1^\infty$ 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.

Standard Derivations & Step-by-Step Problem Solving

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

Memory Aids & JEE Traps

Based on this chapter, here are the top 10 MCQ traps and tricks that appear in JEE:

1. Misconception → Direct substitution in $0/0$ forms gives $0$ or $\infty$. Correct Understanding → $0/0$ is an indeterminate form. You must cancel the vanishing factor (e.g., $(x-a)$) from the numerator and denominator, apply L'Hôpital's rule, or use series expansions. → [JEE TIP] Trap 1 - Phantom Zeroes.
2. Misconception → $\lim_{x \to 0} \frac{\sin(kx)}{x} = 1$. Correct Understanding → $\lim_{x \to 0} \frac{\sin(kx)}{x} = k$. You must multiply and divide by $k$ to apply the standard identity $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$. → [JEE TIP] Trap 2 - Coefficient Missing.
3. Misconception → Trigonometric limit identities work in degrees. Correct Understanding → $\lim_{x \to 0} \frac{\sin x}{x} = 1$ is ONLY valid when $x$ is in radians. For $x$ in degrees, the limit is $\frac{\pi}{180}$. → [JEE TIP] Trap 3 - Degree Trap.
4. Misconception → $\lim_{x \to 0} [x] = 0$ (where $[x]$ is the greatest integer function). Correct Understanding → The limit does NOT exist. LHL = $\lim_{x \to 0^-} [x] = -1$, while RHL = $\lim_{x \to 0^+} [x] = 0$. → [JEE TIP] Trap 4 - GIF Discontinuity.
5. Misconception → When taking limits to infinity involving $\sqrt{x^2}$, you can just write $x$. Correct Understanding → $\sqrt{x^2} = |x|$. If $x \to -\infty$, then $\sqrt{x^2} = -x$. Missing this minus sign ruins the whole problem. → [JEE TIP] Trap 5 - Modulus Trap.
6. Misconception → Sandwich theorem applies if $f(x) \leq g(x) \leq h(x)$ at exactly one point. Correct Understanding → The inequality must hold for an entire neighborhood around the limit point (except possibly at the point itself). → [JEE TIP] Trap 6 - Local Neighborhoods.
7. Misconception → In the quotient rule, $(u/v)' = \frac{u'v + uv'}{v^2}$. Correct Understanding → The sign is negative: $\frac{u'v - uv'}{v^2}$. Always start with the derivative of the numerator. → [JEE TIP] Trap 7 - Sign Error in Division.
8. Misconception → If $\lim f(x)$ does not exist and $\lim g(x)$ does not exist, then $\lim [f(x) + g(x)]$ does not exist. Correct Understanding → The sum can easily have a limit. Example: $f(x) = [x]$ and $g(x) = -[x]$ at integers. → [JEE TIP] Trap 8 - Combining DNE limits.
9. Misconception → The derivative of a function at $x=a$ is always defined if the function is continuous. Correct Understanding → Continuous functions can have sharp corners (like $|x|$ at $x=0$) where LHD $\neq$ RHD, meaning the derivative does not exist. → [JEE TIP] Trap 9 - Sharp Corners.
10. Misconception → $\lim_{x \to 0} \frac{1-\cos x}{x^2} = 0$. Correct Understanding → $\lim_{x \to 0} \frac{1-\cos x}{x} = 0$, but $\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}$ (derived using the half-angle formula $1-\cos x = 2\sin^2(x/2)$). → [JEE TIP] Trap 10 - Squared Denominator in Cosine Limits.