Limits & Continuity – Study Notes
Program: ITI Admission Test
Subject: Mathematics
Topic: Limits & Continuity
1. Simple Explanation
| Concept | What it means | Why it matters |
|---|---|---|
| Limit | The value a function (f(x)) approaches as (x) gets closer to a point (a). | Helps predict behaviour near points where the function may not be defined. |
| Continuity | A function is continuous at (a) if the limit of (f(x)) as (x\to a) equals the actual value (f(a)). | Guarantees “no jumps” or “holes”; essential for calculus and real‑world modelling. |
| One‑sided limits | (\displaystyle \lim_{x\to a^-} f(x)) (from the left) and (\displaystyle \lim_{x\to a^+} f(x)) (from the right). | Needed when a function behaves differently on each side (e.g., piecewise or at a boundary). |
| Infinite limits | When (f(x)) grows without bound as (x\to a). | Describes vertical asymptotes. |
| Limits at infinity | Behaviour of (f(x)) as (x\to \pm\infty). | Describes horizontal asymptotes. |
2. Key Points
2.1 Basic Limit Laws
| Property | Notation | Example |
|---|---|---|
| Sum | (\displaystyle \lim_{x\to a}[f(x)+g(x)]=\lim f(x)+\lim g(x)) | (\lim_{x\to2}(x+3)+(4x-1)=5+7=12) |
| Difference | (\displaystyle \lim[f(x)-g(x)]=\lim f(x)-\lim g(x)) | |
| Product | (\displaystyle \lim[f(x)g(x)]=(\lim f)(\lim g)) | |
| Quotient | (\displaystyle \lim\frac{f}{g}=\frac{\lim f}{\lim g}) ((\lim g\neq0)) | |
| Power | (\displaystyle \lim[f(x)]^n=[\lim f]^n) | |
| Constant multiple | (\displaystyle \lim[c,f(x)]=c,\lim f(x)) |
2.2 Special Limits
| Limit | Result | Reason |
|---|---|---|
| (\displaystyle \lim_{x\to a}\frac{x-a}{x-a}) | 1 | Cancels the factor (except at (x=a)). |
| (\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1) | 1 | Key trigonometric limit. |
| (\displaystyle \lim_{x\to0}\frac{1-\cos x}{x^2}=\tfrac12) | ½ | From Taylor expansion. |
| (\displaystyle \lim_{x\to0}\frac{e^x-1}{x}=1) | 1 | Exponential limit. |
2.3 Continuity
A function (f) is continuous at (a) iff
- (f(a)) is defined.
- (\displaystyle \lim_{x\to a} f(x)) exists.
- (\displaystyle \lim_{x\to a} f(x)=f(a)).
Types of Discontinuities
| Type | Description | Example |
|---|---|---|
| Removable | Limit exists but (f(a)) not equal to it (or not defined). | (f(x)=\frac{x^2-1}{x-1}) at (x=1). |
| Jump | Left and right limits exist but are unequal. | (f(x)=\begin{cases}1,&x<0\2,&x\ge0\end{cases}). |
| Infinite | Limit diverges to (\pm\infty). | (\frac1{x}) at (x=0). |
| Oscillatory | Limit does not exist because values oscillate infinitely. | (\sin\frac1x) as (x\to0). |
2.4 Important Theorems
- Intermediate Value Theorem (IVT): If (f) is continuous on ([a,b]) and (N) is between (f(a)) and (f(b)), then ∃ (c∈(a,b)) with (f(c)=N).
- Squeeze (Sandwich) Theorem: If (g(x)≤f(x)≤h(x)) and (\lim g=\lim h=L), then (\lim f=L).
- L’Hôpital’s Rule (for 0/0 or ∞/∞ indeterminate forms):
[ \lim_{x\to a}\frac{f(x)}{g(x)}= \lim_{x\to a}\frac{f'(x)}{g'(x)}\quad(\text{if right‑hand side exists}) ]
3. Worked Example
Problem
Find
[
\lim_{x\to1}\frac{x^2-3x+2}{x-1}.
]
Solution
- Factor the numerator
[ x^2-3x+2=(x-1)(x-2). ] - Simplify the fraction
[ \frac{(x-1)(x-2)}{x-1}=x-2,\quad \text{for }x\neq1. ] - Take the limit of the simplified expression
[ \lim_{x\to1}(x-2)=1-2=-1.