Functions and Limits

Introduction to Functions

A function $f$ is a correspondence between two sets, $X$ and $Y$, such that each element in $X$ (the domain) is assigned to a unique element in $Y$ (the range or codomain). In board exams, you will frequently be tested on identifying the domain and range of algebraic, trigonometric, and logarithmic functions, as well as finding composite functions $(f \circ g)(x)$ and inverse functions $f^{-1}(x)$. Master the classification of functions—specifically even and odd functions (tested via $f(-x) = f(x)$ and $f(-x) = -f(x)$ respectively) and sketch-based problems involving piecewise-defined functions.

Limits of Functions

The concept of a limit, written as $\lim_{x \to a} f(x) = L$, describes the behavior of a function as the independent variable approaches a specific value, rather than at that value. To score full marks in numerical limit problems, you must be well-versed in resolving indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ using algebraic factorization, rationalization, and standard trigonometric limits. Two critical theorems frequently used in subjective and objective sections are:

1. $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (where $x$ is in radians)
2. $\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$ or $\lim_{n \to \infty} (1 + \frac{1}{n})^n = e$

Continuity and Differentiability

A function $f(x)$ is defined as continuous at a point $x = c$ if three strict conditions are met: $f(c)$ must be defined, $\lim_{x \to c} f(x)$ must exist, and the limit must equal the functional value ($\lim_{x \to c} f(x) = f(c)$). Board exam questions often provide piecewise functions containing unknown constants (like $k$ or $c$) and ask you to find their values by enforcing continuity at a given point. Understanding these foundational concepts of limits and continuity is vital, as they directly bridge the gap into differential calculus (derivatives and first-principle proofs).