Exhaustive Guide to Functions and Limits for Aptitude and Intermediate Studies

Introduction to the Lecture Series by Mohammad Hashim Zia

Instructor Context: This lecture is part of an exhaustive series by Mohammad Hashim Zia (Math Flix by Hashim Zia), designed for university aptitude tests (NET for NUST, FAST, NED, IBA, UET, COMSATS, PIEAS, GIKI, Mehran) and Intermediate college exams.
Goal: The series provides a definitive guide for achieving $100\, \text{out of}\, 100$ in MCQ sections, combining conceptual overviews with rapid-solving techniques (shortcuts).
Methodology: Each session is divided into a conceptual summary followed by the resolution of over $80$ specific MCQs from past papers and textbooks.

Fundamental Concepts of Functions

Definition of a Function: In mathematics, a function is a specific type of relation between an input and an output. It assigns each element $x$ in set $X$ to a unique element $y$ in set $Y$ .
Functional Notation: Represented as $y = f(x)$ .
- $x$ is the input (independent variable).
- $y$ is the output (dependent variable).
The Unique Output Rule: A relation is only a function if every input produces exactly one output.
- Multiple Inputs to One Output: Permissible (e.g., several switches controlling one light). Case: $f(1) = a$ and $f(2) = a$ is a valid function.
- One Input to Multiple Outputs: Prohibited. Case: $f(1) = a$ and $f(1) = b$ is not a function.
- Input Left Out: In set notation, every element in the domain must be mapped. If an input has no output, it is not a function.
- Output Left Out: Elements in the codomain do not necessarily need to be mapped; this depends on the function type (Into vs. Onto).
Naming Conventions: Functions are also extensively referred to as "Mapping" in academic contexts.

Domain, Range, and Codomain

Domain: The set of all possible input values ( $x$ values) for which the function is defined.
Range: The set of actual output values ( $y$ values) that result from the mapping.
Codomain: The set of all potentially possible output values (expected results), which may include the Range as a subset.
Numerical Examples:
- If $f(x) = x^2$ :
  - Domain $2 \rightarrow$ Range $4$ .
  - Domain $3 \rightarrow$ Range $9$ .
  - Domain $4 \rightarrow$ Range $16$ .
Comparison: If a codomain is $\{a, b, c, d\}$ but only $\{a, b, c\}$ are achieved as results, the Range is $\{a, b, c\}$ .

Types of Functions and Mappings

Injective (One-to-One) Function: Each unique domain element maps to a unique range element. No two different inputs share the same output.
Surjective (Onto) Function: Every element in the codomain is mapped. Mathematically, $\text{Codomain} = \text{Range}$ .
Bijective Function: A function that is both Injective and Surjective. It features perfectly unique one-to-one mapping where no outputs are left unused. This is required for a function to have an inverse.
Into Function: A function where the Codomain is larger than the Range ( $\text{Codomain} \neq \text{Range}$ ), meaning some outputs are left unmapped.
Constant Function: A function where every input maps to the same single constant value $c$ . (e.g., $f(x) = 7$ for all $x$ ).
Identity Function: A function where the output is identical to the input ( $f(x) = x$ ). Example: $f(4) = 4$ . The domain and range are identical.

Inverse of a Function

Concept: Inverse mapping ( $f^{-1}(x)$ ) essentially reverses the direction of the function (mapping $Y$ back to $X$).
Three-Step Procedure to Find Inverse:
1. Replace $f(x)$ with $y$ .
2. Isolate $x$ algebraically to express the function in terms of $y$ .
3. Interchange $x$ with $f^{-1}(x)$ and $y$ with $x$ .
Advanced Example (NUST Paper MCQ):
- Given $f(x) = \frac{x - 3}{x + 6}$ .
- Let $y = \frac{x - 3}{x + 6} \implies y(x + 6) = x - 3 \implies yx + 6y = x - 3$ .
- $yx - x = -6y - 3 \implies x(y - 1) = -(6y + 3) \implies x = \frac{-(6y + 3)}{y - 1}$ .
- Inverse: $f^{-1}(x) = \frac{-(6x + 3)}{x - 1}$ .

Composite Functions

Definition: A function followed by another function, expressed as $f(g(x))$ (often written as $f \circ g$ ) or $g(f(x))$ ( $g \circ f$ ).
Conceptual Application: The interior function acts as the input/domain for the exterior function.
Multi-layered Composition: Functions can be nested deeper, such as $g(f(h(x)))$ . Always solve from the innermost layer outward.

Even and Odd Functions

Even Functions:
- Condition: $f(-x) = f(x)$ .
- Symmetry: Symmetric about the $y$ -axis.
- Trigonometric Example: $\cos(-x) = \cos(x)$ .
Odd Functions:
- Condition: $f(-x) = -f(x)$ .
- Symmetry: Symmetric about the Origin.
- Trigonometric Example: $\sin(-x) = -\sin(x)$ and $\tan(-x) = -\tan(x)$ .
Neither Case: If a function satisfies neither condition upon substituting $-x$ , it is classified as "Neither Even Nor Odd."

Introduction to Limits

Definition: A limit describes the behavior of a function at or near a specific point $x$ . It is written as $\lim_{x \to a} f(x)$ .
Existence of Limits: A limit exists only if the Left Hand Limit (LHL) equals the Right Hand Limit (RHL).
Convergence and Divergence:
- Convergent Series: A sequence or series that approaches a specific fixed finite number as its limit.
- Divergent Series: A sequence that does not approach a finite number (approaches infinity or oscillates).
- Aptitude Test Trick: For a series like $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$ , as $n \to \infty$ , the denominator increases, making the term approach $0$ . This is a convergent series.

Numerical and Structural Properties of Limits

Constant Property: $\lim_{x \to a} [c] = c$ (Limit of a constant is the constant itself).
Scalar Multiplication: $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$ .
Arithmetic Operations: Limits distribute over addition, subtraction, multiplication, and division.
- $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$ .
- $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$ .
Power Rule: $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$ .

L'Hôpital's Rule (L'Hospital Rule)

Application: A shortcut for solving indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .
Method: Differentiate the numerator and denominator separately until the function is defined upon substitution of the limit value.
Mathematical Warning: This rule is strictly for MCQs and Aptitude tests. Using it in Board exams for high-mark subjective questions may lead to zero marks as examiners require standard algebraic methods.
Example: $\lim_{x \to 0} \frac{1 - \cos(7x)}{1 - \cos(5x)}$ .
- First derivative: $\frac{7 \sin(7x)}{5 \sin(5x)}$ (still $\frac{0}{0}$ ).
- Second derivative: $\frac{49 \cos(7x)}{25 \cos(5x)}$ .
- Substitution: Since $\cos(0) = 1$ , the answer is $\frac{49}{25}$ .

Infinite Limit Shortcuts for Rational Functions

Context: Solving $\lim_{x \to \infty} \frac{f(x)}{g(x)}$ where $f(x)$ and $g(x)$ are algebraic.
The Leading Coefficient Rule:
- Identify the highest power of $x$ in the entire fraction.
- If the degrees of the numerator and denominator are equal, the limit is the ratio of their coefficients. Example: $\lim_{x \to \infty} \frac{3x^2 + 5}{2x^2 - 1} = \frac{3}{2}$ .
- If the numerator has a higher degree, the limit is $\infty$ .
- If the denominator has a higher degree, the limit is $0$ .

Fundamental Limit Formulas to Memorize

$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
$\lim_{x \to \infty} \frac{1}{x} = 0$
$\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$
$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$
$\lim_{n \to \infty} \frac{\ln(e^x + 1)}{x} = 1$
$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n \times a^{n-1}$
$\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)$
$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ (for $x$ in radians).
$\lim_{x \to 0} \frac{\tan(x)}{x} = 1$

Hyperbolic and Inverse Hyperbolic Functions

Definitions:
- $\sinh(x) = \frac{e^x - e^{-x}}{2}$
- $\cosh(x) = \frac{e^x + e^{-x}}{2}$
- $\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$
Inverse Hyperbolic Identities:
- $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 + 1})$
- $\cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1})$
- $\tanh^{-1}(x) = \frac{1}{2} \ln(\frac{1 + x}{1 - x})$
- $\text{sech}^{-1}(x) = \ln(\frac{1 + \sqrt{1 - x^2}}{x})$
- $\text{csch}^{-1}(x) = \ln(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|})$
- $\coth^{-1}(x) = \frac{1}{2} \ln(\frac{x + 1}{x - 1})$

Continuity and Special Cases

Continuity Conditions: A function $f(x)$ is continuous at $x = a$ if:
1. $f(a)$ is defined (exists).
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$ .
Function within a Specific Domain: If a question provides a domain like $[-1, 5]$ and asks for a value at $x = -3$ , the answer is "Does Not Exist" (DNE) because the point is outside the prescribed domain.
Trigonometric Limits with Degrees: If the limit involves degrees ( $x^\circ$ ), you must convert to radians using $x^\circ = \frac{\pi x}{180}$ before applying $\frac{\sin(x)}{x} = 1$ . The result is often $\frac{\pi}{180}$ .

Questions & Discussion

Aptitude Coaching Mafia: Zia discusses the commercialization of education, noting that many large institutes (10-15 campuses) fail to produce top scorers despite high fees ( $30,000$ - $40,000$ PKR). He highlights his student, Rameez Kamran, who achieved high marks ( $96$ ) through online preparation.
Physical vs. Online Learning: Zia disputes the idea that physical institutes are superior, citing that top results are now coming from focused online curricula.
Board Exam Calculation: Zia confirms that calculators are only allowed in NED and PIEAS entry tests, whereas most other universities prohibit them, requiring students to know trigonometric values (e.g., $\cos(0)=1, \sin(90)=1$ ) from memory.