Limits & Continuity – Study Notes
(For PSU / GATE Mathematics)
Aim – Understand the formal definition of limits, continuity, and common techniques to evaluate them.
1. Simple Explanation
| Concept | Intuition | Formal Statement |
|---|---|---|
| Limit | As (x) gets arbitrarily close to (a), the function (f(x)) approaches a value (L). | (\displaystyle \lim_{x\to a} f(x)=L) iff for every (\varepsilon>0) there exists (\delta>0) such that [0< |
| Continuity at (a) | No “jump” or “hole” at (x=a); the function value equals the limit. | (f) is continuous at (a) iff (\displaystyle \lim_{x\to a} f(x)=f(a).) |
| Right / Left limits | Approach from one side only. | (\displaystyle \lim_{x\to a^+} f(x)=L^+), (\displaystyle \lim_{x\to a^-} f(x)=L^-). |
| Discontinuity types | “Removable”, “Jump”, “Infinite”, “Oscillatory”. |
2. Key Points
| Topic | Key Points |
|---|---|
| Existence of Limits | 1. If (\lim_{x\to a}f(x)) exists, both one‑sided limits exist and are equal. 2. If either one‑sided limit diverges, the two‑sided limit does not exist. |
| Limit Laws | (i) (\lim (f+g)=\lim f+\lim g) (ii) (\lim (fg)=\lim f\cdot\lim g) (iii) (\lim\frac{f}{g}=\frac{\lim f}{\lim g}) (provided (\lim g\neq0)) (iv) (\lim c f = c\lim f) |
| Continuity Properties | 1. Sum, product, quotient (with non‑zero denominator) of continuous functions are continuous. 2. Composition (g\circ f) is continuous if (f) is continuous at (a) and (g) is continuous at (f(a)). |
| Piecewise Functions | Check continuity at boundary points by verifying left/right limits and the function value. |
| Removable Discontinuity | Occurs when (\lim_{x\to a}f(x)) exists but (f(a)\neq \lim f). Can be “fixed” by redefining (f(a)). |
| Jump Discontinuity | Left and right limits exist but are unequal. |
| Infinite Discontinuity | At least one one‑sided limit is (\pm\infty). |
| Oscillatory Discontinuity | Limits from either side do not exist due to infinite oscillations. |
| Continuity on Intervals | A function is continuous on ([a,b]) iff it is continuous at every point in ((a,b)) and has one‑sided limits at (a,b). |
| Important Theorems | Intermediate Value Theorem, Bolzano’s theorem, Weierstrass theorem – useful for existence proofs. |
3. Worked Example
Problem
Evaluate
[ \lim_{x\to 0} \frac{\sin 3x}{x} ]
and determine whether the function
[ f(x)= \begin{cases} \dfrac{\sin 3x}{x}, & x\neq 0 \ k, & x=0 \end{cases} ]
is continuous at (x=0). If it is not continuous, find the value of (k) that makes it continuous.
Step 1: Use a Standard Limit
Recall the fundamental limit
[ \lim_{u\to 0}\frac{\sin u}{u}=1 . ]
Let (u = 3x). When (x\to 0), (u\to 0), and
[ \frac{\sin 3x}{x} = \frac{\sin u}{u}\cdot 3 ]
since (x = u/3). Therefore,
[ \lim_{x\to 0}\frac{\sin 3x}{x} = \lim_{u\to 0}\left( \frac{\sin u}{u}\right)\cdot 3 = 1\cdot 3 = 3. ]
Step 2: Continuity at (x=0)
For continuity at (0),
[ \lim_{x\to 0}f(x) = f(0). ]
We found (\displaystyle \lim_{x\to 0}f(x) = 3).
Hence (f) is continuous at (0) iff (k=3).
- If (k=3): (f(0)=3) and (\lim_{x\to0}f(x)=3) → continuous.
- If (k\neq 3): mismatch → discontinuity (removable).
Step 3: Classify the Discontinuity (if (k\neq3))
The function has a removable discontinuity at (x=0) because:
- Both one‑sided limits exist (equal to 3).
- The function value is not equal to this limit.
4. Quick Revision Summary
| Topic | What to Recall |
|---|---|
| Limit Definition | Epsilon–delta condition. |
| Limit Laws | Sum, product, quotient, scalar multiple. |
| Continuity at a Point | (\lim f = f(a)). |
| Piecewise Continuity | Check left/right limits at boundaries. |
| Discontinuity Types | Removable, jump, infinite, oscillatory. |
| Continuity on Intervals |