2. Calculus — Differential Calculus

Limit of a function

The limit of a function forms the foundational bedrock of all differential and integral calculus, capturing the precise behavior of a function as its independent variable approaches a specific point.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Cauchy's / $\epsilon$-$\delta$ Definition of Limit

Let $f: A \subset \mathbb{R} \to \mathbb{R}$ and let $c$ be a limit point of $A$. We say that $\lim_{x \to c} f(x) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x \in A$, if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$.

📐 Geometric Interpretation

Geometrically, as $x$ is chosen within a horizontal strip of radius $\delta$ around $c$ (excluding $c$ itself), the corresponding function values $f(x)$ are forced into a vertical strip of radius $\epsilon$ around $L$.

⚙️ Algebraic & Structural Meaning

Establishes the rigorous topological foundation of real analysis, guaranteeing that algebraic limit laws (sum, product, quotient rules) hold securely without circular reasoning.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Existence of Limit Point	`Domain Neighborhood $(c-\delta, c+\delta) \setminus \{c\}$`	Strictly Mandatory / Necessary	If $c$ is an isolated point of the domain, the limit concept becomes vacuous or undefined.
Equality of One-Sided Limits	`Left interval $(c-\delta, c)$ and Right interval $(c, c+\delta)$`	Strictly Mandatory	If $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$, the two-sided limit $\lim_{x \to c} f(x)$ fails to exist.
Finiteness of Limit Value	`Real number codomain $L \in \mathbb{R}$`	Essential for standard limit existence	If values grow unboundedly (diverge to $\pm\infty$), the limit does not exist in the real number system.

Useful results and shortcuts

Use these identities and results when their conditions apply:

L'Hôpital's Rule for $\frac{0}{0}$ or $\frac{\infty}{\infty}$

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$

Rule: Directly differentiate numerator and denominator separately until the indeterminate form is resolved, provided the quotient of derivatives exists.

Application: Directly applied in over 40% of UP PGT/TGT calculus limit problems.

Standard Algebraic Limits Form

$$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$$

Rule: Instantly substitute $n$ and $a$ when encountering binomial rational limits, saving algebraic factorization time.

Exponential / Logarithmic Standard Limits

$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$$

Rule: Memorize standard exponential and logarithmic limits to bypass full series expansions during tight exam time constraints.

Trigonometric Ratio Limits

$$\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$$

Rule: Replace $\sin(kx) \approx kx$ and $1 - \cos(kx) \approx \frac{1}{2}k^2x^2$ for rapid asymptotic approximations.

Application: Essential speed tool for multi-step calculus problems.

The $1^\infty$ Indeterminate Form Shortcut

$$\text{If } \lim f(x) = 1 \text{ and } \lim g(x) = \infty, \text{ then } \lim [f(x)]^{g(x)} = e^{\lim (f(x)-1)g(x)}$$

Rule: Convert $1^\infty$ problems into exponential products instantly without cumbersome logarithmic transformations.

Application: Frequently featured as a high-difficulty differentiator in UP PGT.

Series Expansion (Maclaurin) Elimination

$$\lim_{x \to 0} \frac{x - \sin x}{x^3} = \frac{1}{6}$$

Rule: Use leading non-zero terms of Maclaurin series for rational functions involving trigonometric or exponential terms.

Application: Enables elimination of incorrect options.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Misapplying L'Hôpital's Rule on Non-Indeterminate Forms

Common Pitfall: Candidates blindly differentiate numerator and denominator when the expression is not in $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form.

Counterexample: $$\lim_{x \to 0} \frac{\sin x}{x + 1} = \frac{0}{1} = 0$$

Why it fails: Applying L'Hôpital gives $\lim \frac{\cos x}{1} = 1$, which is completely incorrect because direct substitution yields $0$.

🛡️ Exam Defense: Always evaluate by direct substitution first before applying any limit rule.

⚠️ COMMON ERROR #02

Ignoring Left-Hand and Right-Hand Limit Discrepancies

Common Pitfall: Assuming a limit exists simply because algebraic manipulation yields a single formula, ignoring absolute values or step functions.

Counterexample: $$\lim_{x \to 0} \frac{|x|}{x}$$

Why it fails: Left-hand limit is $-1$ and right-hand limit is $+1$, hence the two-sided limit does not exist.

🛡️ Exam Defense: Check one-sided limits for modulus, greatest integer, or signum functions.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Algebraic Limits	Functions defined in neighborhood	$\lim (f \pm g) = \lim f \pm \lim g$	Combined ordinate approach	Polynomial or rational functions
Sandwich Theorem	$g(x) \le f(x) \le h(x)$	If $\lim g = \lim h = L$, then $\lim f = L$	Function trapped between two bounding curves	Trigonometric inequalities with ratios
L'Hôpital's Rule	$\frac{0}{0}$ or $\frac{\infty}{\infty}$, differentiable	$\lim \frac{f}{g} = \lim \frac{f'}{g'}$	Ratio of tangents at the limiting point	Complex algebraic/trig quotients
One-Sided Limits	Domain restricted to one side	$\lim_{x \to c^-} f(x) = L_1$, $\lim_{x \to c^+} f(x) = L_2$	Approach along left or right curve branch	Modulus, greatest integer, piecewise functions

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Evaluate the limit: $$\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$$

(A) $0$(B) $\frac{1}{2}$(C) $1$(D) $\frac{1}{3}$

📚 Method A: Formal Textbook Derivation

Write $\tan x = x + \frac{x^3}{3} + \dots$ and $\sin x = x - \frac{x^3}{6} + \dots$. Then numerator is $\left(x + \frac{x^3}{3}\right) - \left(x - \frac{x^3}{6}\right) = \frac{x^3}{2}$. Dividing by $x^3$ gives $\frac{1}{2}$.

⚡ Method B: 25-Second Exam Shortcut

Apply L'Hôpital's rule three times successively or use standard series expansion.

💡 Tip: Series expansion is faster than three rounds of L'Hôpital's rule.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Find the value of $k$ if $f(x) = \frac{\sqrt{1+kx} - \sqrt{1-kx}}{x}$ has a limit of $2$ as $x \to 0$.

(A) $1$(B) $2$(C) $3$(D) $4$

📚 Method A: Formal Textbook Derivation

Rationalize the numerator by multiplying by $(\sqrt{1+kx} + \sqrt{1-kx})$. This simplifies to $\frac{2kx}{x(\sqrt{1+kx} + \sqrt{1-kx})} = \frac{2k}{\sqrt{1+kx} + \sqrt{1-kx}}$. Taking $x \to 0$ gives $\frac{2k}{2} = k$. Given limit is $2$, so $k = 2$.

⚡ Method B: 25-Second Exam Shortcut

Use binomial approximation $(1+u)^n \approx 1 + nu$: numerator becomes $1 + \frac{1}{2}kx - (1 - \frac{1}{2}kx) = kx$. Thus limit is $k = 2$.

💡 Tip: Binomial approximation cuts down algebraic steps by 70%.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: Evaluate $\lim_{x \to 0} \frac{\sin(ax)}{\sin(bx)}$.

A $\frac{a}{b}$

B $\frac{b}{a}$

C $1$

D $ab$

Derivation & Analysis:

Rewrite as $\lim_{x \to 0} \frac{\sin(ax)}{ax} \cdot \frac{bx}{\sin(bx)} \cdot \frac{a}{b} = 1 \cdot 1 \cdot \frac{a}{b} = \frac{a}{b}$.

MCQ 2: Evaluate $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{2x}$.

A $e$

B $e^2$

C $1$

D $\infty$

Derivation & Analysis:

Using standard limit $\lim_{x \to \infty} (1 + \frac{a}{x})^{bx} = e^{ab}$, here $a=1, b=2$, so result is $e^{1 \cdot 2} = e^2$.

MCQ 3: The limit $\lim_{x \to 0} x \sin\left(\frac{1}{x}\right)$ is equal to:

A $1$

B $-1$

C $0$

D Does not exist

Derivation & Analysis:

By Sandwich Theorem, $-|x| \le x \sin(1/x) \le |x|$. Since $\lim_{x \to 0} -|x| = \lim_{x \to 0} |x| = 0$, the limit is $0$.

MCQ 4: Evaluate $\lim_{x \to 0} \frac{a^x - 1}{x}$ ($a > 0$).

A $1$

B $a$

C $\ln a$

D $0$

Derivation & Analysis:

This is the standard definition of derivative of $a^x$ at $x=0$, which evaluates directly to $\ln a$.

MCQ 5: If $f(x) = \frac{|x|}{x} for x \neq 0$, then $\lim_{x \to 0^+} f(x) - \lim_{x \to 0^-} f(x)$ is:

A $0$

B $1$

C $-2$

D $2$

Derivation & Analysis:

Right-hand limit is $+1$ and left-hand limit is $-1$. Difference is $1 - (-1) = 2$.

MCQ 6: Evaluate $\lim_{x \to 0} \frac{\log(1+x)}{x}$.

A $0$

B $1$

C $e$

D $\infty$

Derivation & Analysis:

Standard limit formula yielding $1$.

MCQ 7: Evaluate $\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$.

A $2$

B $3$

C $4$

D $6$

Derivation & Analysis:

Factorize numerator: $(x-2)(x^2+2x+4)$ and denominator: $(x-2)(x+2)$. Cancel $(x-2)$ and substitute $x=2$ to get $\frac{4+4+4}{4} = 3$.

MCQ 8: Which of the following limits does NOT exist?

A $\lim_{x \to 0} \frac{\sin x}{x}$

B $\lim_{x \to 0} e^{1/x}$

C $\lim_{x \to 0} x^2 \cos(1/x)$

D $\lim_{x \to 1} \frac{x^2-1}{x-1}$

Derivation & Analysis:

For $\lim_{x \to 0} e^{1/x}$, the left-hand limit is $0$ and right-hand limit is $\infty$, so it does not exist.

MCQ 9: Evaluate $\lim_{x \to 0} \frac{1 - \cos(2x)}{x^2}$.

A $1$

B $2$

C $4$

D $0$

Derivation & Analysis:

Since $1 - \cos(2x) = 2\sin^2 x$, expression becomes $\lim \frac{2\sin^2 x}{x^2} = 2(1)^2 = 2$.

Q10

MCQ 10: Find $\lim_{x \to \infty} \left(\frac{x+2}{x-1}\right)^x$.

A $e$

B $e^2$

C $e^3$

D $1$

Derivation & Analysis:

Rewrite as $\left(1 + \frac{3}{x-1}\right)^x = \left(1 + \frac{3}{x-1}\right)^{x-1} \cdot \left(1 + \frac{3}{x-1}\right)^1 \to e^3 \cdot 1 = e^3$.

Common questions

Not always. For standard algebraic, trigonometric, and exponential functions within their continuous domain, direct substitution is valid. One-sided limits are mandatory only for piecewise functions, modulus functions, greatest integer functions, or points where the function definition splits.

Applying L'Hôpital's rule prematurely without verifying that the expression is in an indeterminate form (like $\frac{0}{0}$ or $\frac{\infty}{\infty}$) is the leading cause of avoidable errors.

Master standard expansions (Maclaurin series), algebraic factorization shortcuts, and asymptotic approximations (like $\sin x \approx x$) to solve complex problems.

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Differentiation of logarithmic functions