Limits & Continuity
State Polytechnic Entrance – Mathematics
Goal: Understand the formal definition of limits, how to evaluate them, and the concept of continuity.
Why it matters: Limits form the foundation for derivatives and integrals. Continuity tells you whether a function behaves “nicely” without jumps, holes or asymptotes.
1. Simple Explanation
| Concept | What it means | Why it matters |
|---|---|---|
| Limit | The value a function approaches as the input approaches a particular point. | Gives us a precise way to talk about behavior near points where the function may not be defined. |
| Continuity | A function is continuous at a point if the limit there equals the function’s value. | Continuous functions can be graphed without breaks, essential for calculus. |
2. Key Points
2.1 Formal Definition of Limit
For a real‑valued function (f(x)) and a real number (a):
[ \lim_{x\to a} f(x) = L ]
iff for every (\epsilon>0) there exists (\delta>0) such that
[ 0<|x-a|<\delta \quad\Rightarrow\quad |f(x)-L|<\epsilon . ]
- (\epsilon): how close (f(x)) must be to (L).
- (\delta): how close (x) must be to (a).
2.2 Types of Limits
| Type | Symbol | When it occurs |
|---|---|---|
| Finite limit | (\displaystyle\lim_{x\to a} f(x)=L) | (f(x)) approaches a real number. |
| One‑sided limit | (\displaystyle\lim_{x\to a^-} f(x),\ \lim_{x\to a^+} f(x)) | Approach from left/right. |
| Infinite limit | (\displaystyle\lim_{x\to a} f(x)=\pm\infty) | (f(x)) grows without bound. |
| Limit at infinity | (\displaystyle\lim_{x\to\infty} f(x)) | Behaviour as (x) grows large. |
| Limit of a sequence | (\displaystyle\lim_{n\to\infty} a_n) | Same idea for sequences. |
2.3 Rules for Calculating Limits
| Rule | Statement | Example |
|---|---|---|
| Algebraic | (\lim (f\pm g) = \lim f \pm \lim g) | (\lim (x+3)=4) as (x\to1). |
| Product | (\lim (f\cdot g)= (\lim f)(\lim g)) | (\lim (2x\cdot x)=\lim 2x^2 = 8) as (x\to2). |
| Quotient | (\lim (f/g) = (\lim f)/(\lim g)), provided (\lim g\neq0). | (\lim (x^2-1)/(x-1)=\lim (x+1)=2) as (x\to1). |
| Squeeze | If (g(x)\le f(x)\le h(x)) and (\lim g=\lim h=L), then (\lim f=L). | (\lim x^2\sin(1/x)=0) as (x\to0). |
| Change of Variable | Substitute (u=g(x)) if (\lim g(x)=a). | (\lim_{x\to0}\frac{\sin x}{x}=1). |
| L’Hôpital’s Rule | If (\frac{0}{0}) or (\frac{\infty}{\infty}), then (\lim f/g = \lim f'/g'). | (\lim_{x\to0}\frac{\sin x}{x}=1). |
2.4 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)).
Common continuous functions
| Function | Domain | Continuous? |
|---|---|---|
| Polynomial | (\mathbb{R}) | Yes |
| Rational (\frac{p(x)}{q(x)}) | (q(x)\neq0) | Yes where defined |
| Exponential (e^x), (\ln x) | (x>0) | Yes |
| Trig (\sin x, \cos x) | (\mathbb{R}) | Yes |
| Piecewise | Depends on matching values | Check each point |
2.5 Important Theorems
| Theorem | Statement |
|---|---|
| Intermediate Value Theorem | If (f) is continuous on ([a,b]) and (N) between (f(a)) and (f(b)), then (\exists c\in(a,b)) with (f(c)=N). |
| Composition | If (g) is continuous at (a) and (f) is continuous at (g(a)), then (f\circ g) is continuous at (a). |
| Uniform Continuity | On a closed interval ([a,b]), continuous functions are uniformly continuous. |
3. Worked Example
Problem: Evaluate
[ \lim_{x\to 2}\frac{x^3-8}{x-2} ]
Solution Step‑by‑Step
- Recognize the indeterminate form.
Plugging (x=2) gives \