Functions & Limits – KVPY Study Notes
(Prepared for the Kishore Vaigyanik Protsahan Yojana – “Historic” Program)
1. What is a Limit?
Simple Explanation
A limit describes the value that a function (f(x)) approaches as the input (x) gets arbitrarily close to a specific point (a). It tells us the “trend” of the function near (a) without requiring the function to be defined at (a) itself.
Notation
[ \lim_{x \to a} f(x) = L ] means “as (x) approaches (a), (f(x)) tends to (L).”
2. Key Points
| Concept | What It Means | Formula / Example |
|---|---|---|
| Left‑hand limit | Value approached as (x) approaches (a) from the left (smaller than (a)). | (\displaystyle \lim_{x \to a^-} f(x) = L) |
| Right‑hand limit | Value approached as (x) approaches (a) from the right (larger than (a)). | (\displaystyle \lim_{x \to a^+} f(x) = L) |
| Limit exists | Both one‑sided limits exist and are equal. | (\displaystyle \lim_{x \to a} f(x)=L) iff (\lim_{x \to a^-} f(x)=\lim_{x \to a^+} f(x)=L). |
| Continuity at (a) | (f) defined at (a) and (\displaystyle \lim_{x \to a} f(x)=f(a)). | |
| Limit Laws | Basic algebraic rules for manipulating limits. | <ul><li>Sum: (\lim (f+g)=\lim f + \lim g)</li><li>Product: (\lim (fg)=\lim f \cdot \lim g)</li><li>Quotient: (\lim (f/g)=\frac{\lim f}{\lim g}) if (\lim g\neq0)</li><li>Power/Root: (\lim (f^n)= (\lim f)^n) (for integer (n)), (\lim \sqrt[n]{f}=\sqrt[n]{\lim f})</li></ul> |
| Epsilon‑Delta Definition | Formal definition of limit. | For every (\varepsilon>0) there exists (\delta>0) such that if (0< |
| Squeeze (Sandwich) Theorem | If (g(x)\le f(x)\le h(x)) near (a) and (\lim g=\lim h=L), then (\lim f=L). | |
| Standard Trigonometric Limits | Useful for many problems. | (\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1,\ \lim_{x\to0}\frac{1-\cos x}{x}=0,\ \lim_{x\to0}\frac{\tan x}{x}=1). |
| Indeterminate Forms | Forms like (0/0,\ \infty/\infty,\ 0\cdot\infty,\ \infty-\infty,\ 0^0,\ \infty^0,\ 1^\infty) need special handling. | Use algebra, L’Hospital’s rule, series, or other techniques. |
| L’Hospital’s Rule | If (\lim f=\lim g=0) or (\pm\infty) and (\frac{f'}{g'}) has a limit, then (\lim f/g = \lim f'/g'). | |
| Limits at Infinity | Behaviour as (x\to\pm\infty). | (\displaystyle \lim_{x\to\infty} f(x)=L) means (f(x)) gets arbitrarily close to (L) as (x) grows large. |
| Infinite Limits | (\displaystyle \lim_{x\to a} f(x)=\pm\infty). | |
| Limits of Sequences | Special case where (x=n) is integer. | (\displaystyle \lim_{n\to\infty} a_n = L). |