Limits & Continuity for Multivariable Functions

🎯 Learning Objectives

• Evaluate limits of multivariable functions along different paths
• Determine when a limit does not exist using path dependence
• Use the epsilon-delta definition of continuity
• Apply the squeeze theorem for multivariable limits
• Understand that continuity in each variable separately does not imply joint continuity

📋 Prerequisites

• Multivariable Functions (this week) — notation for $f: \mathbb{R}^n \to \mathbb{R}$
• Single-variable limits (BSMA1001) — limit definition, squeeze theorem
• Limits are the foundation of calculus; continuity is essential for differentiability

1. Intuition: Approaching a Point from All Directions

2. Limits in $\mathbb{R}^n$

2.1 Definition

Definition (Limit). $\lim_{\mathbf{x} \to \mathbf{a}} f(\mathbf{x}) = L$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that:

2.2 Path Dependence

• $y = 0$ (approach along $x$-axis)
• $x = 0$ (approach along $y$-axis)
• $y = mx$ (approach along any line)
• $y = kx^2$ (approach along parabola)
• $x = 0$, $y = 0$ (coordinate axes)

Example 1: Limit exists

$f(x,y) = \frac{x^2y}{x^2 + y^2}$. Find $\lim_{(x,y)\to(0,0)} f(x,y)$.

All paths give 0. Use squeeze theorem to confirm: $|f(x,y)| = \frac{x^2|y|}{x^2+y^2} \leq |y|$ (since $\frac{x^2}{x^2+y^2} \leq 1$). As $(x,y)\to(0,0)$, $|y|\to0$, so $|f|\to0$. Limit = 0 ✓. Example 2: Limit DNE (path dependence)

$f(x,y) = \frac{xy}{x^2 + y^2}$. Find $\lim_{(x,y)\to(0,0)} f(x,y)$.

Different lines give different limits (0, 1/2, 2/5, ...). Limit DNE.

3. Continuity

3.1 Definition

Definition (Continuity). $f$ is continuous at $\mathbf{a}$ if:

3.2 Properties

• Sums, products, and quotients (where denominator ≠ 0) of continuous functions are continuous
• Compositions of continuous functions are continuous
• Polynomials in several variables are continuous everywhere
• Rational functions are continuous where the denominator ≠ 0

4. The Squeeze Theorem

Theorem (Squeeze). If $|f(\mathbf{x}) - L| \leq g(\mathbf{x})$ near $\mathbf{a}$ and $\lim_{\mathbf{x}\to\mathbf{a}} g(\mathbf{x}) = 0$, then $\lim_{\mathbf{x}\to\mathbf{a}} f(\mathbf{x}) = L$. Example 3: Using squeeze theorem

$f(x,y) = \frac{x^3}{x^2 + y^2}$. Show $\lim_{(x,y)\to(0,0)} f(x,y) = 0$.

$|f(x,y)| = \frac{|x|^3}{x^2+y^2} = |x| \cdot \frac{x^2}{x^2+y^2} \leq |x|$.

Since $|x| \to 0$ as $(x,y)\to(0,0)$, by squeeze theorem $|f| \to 0$. ✓

5. Common Pitfalls

❌ Pitfall 1: Checking only lines

❌ Pitfall 2: Using polar coordinates incorrectly

❌ Pitfall 3: Separate continuity ≠ joint continuity

6. Formula Summary Table

7. 📝 Practice Questions

Q1: Limit along lines

$f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}$. Does $\lim_{(x,y)\to(0,0)} f$ exist?

Solution: Along $y=0$: $f(x,0) = 1 \to 1$. Along $x=0$: $f(0,y) = -1 \to -1$. Different limits ⇒ DNE. Q2: Limit exists

$f(x,y) = \frac{\sin(xy)}{y}$. Does $\lim_{(x,y)\to(0,0)} f$ exist?

Solution: Along $y=0$? Not defined. Use the fact $\sin(u)/u \to 1$. Write $\frac{\sin(xy)}{y} = x \cdot \frac{\sin(xy)}{xy}$. As $(x,y)\to(0,0)$, $xy \to 0$, so $\frac{\sin(xy)}{xy} \to 1$. Therefore $f(x,y) \to 0 \cdot 1 = 0$. Limit = 0. Q3: Limit DNE — parabolic path

$f(x,y) = \frac{x^2y}{x^4 + y^2}$. Show $\lim_{(x,y)\to(0,0)} f$ DNE.

Solution: Along $y = mx$: $f(x,mx) = \frac{mx^3}{x^4 + m^2x^2} = \frac{mx}{x^2 + m^2} \to 0$ for $m \neq 0$. Along $y = x^2$: $f(x,x^2) = \frac{x^2(x^2)}{x^4 + x^4} = \frac{x^4}{2x^4} = \frac{1}{2}$. Different limits (0 vs 1/2) ⇒ DNE. Q4: Squeeze theorem

$f(x,y) = \frac{x^2y^2}{x^2 + y^2}$. Show $\lim f = 0$ at $(0,0)$.

Solution: $0 \leq \frac{x^2y^2}{x^2+y^2} \leq \frac{(x^2+y^2)(x^2+y^2)}{x^2+y^2} = x^2+y^2$ (since $x^2y^2 \leq (x^2+y^2)^2/4$, actually $x^2y^2 \leq (x^2+y^2)^2/4$). Better: $0 \leq \frac{x^2y^2}{x^2+y^2} \leq \frac{x^2(x^2+y^2)}{x^2+y^2} = x^2$ (since $y^2 \leq x^2+y^2$). As $x\to0$, $x^2\to0$, so limit = 0 by squeeze. Q5: Continuity

Is

continuous at $(0,0)$?

Solution: Earlier we showed the limit DNE along different lines. So $\lim f \neq f(0,0)=0$. $f$ is not continuous at $(0,0)$. Q6: Making a function continuous

Define $f(0,0)$ to make $f(x,y) = \frac{x^2y}{x^2+y^2}$ continuous.

Solution: From Example 1, $\lim_{(x,y)\to(0,0)} f = 0$. Define $f(0,0) = 0$. Q7: Polar coordinates

Use polar coordinates to evaluate $\lim_{(x,y)\to(0,0)} \frac{x^3}{x^2+y^2}$.

Solution: $x = r\cos\theta$, $y = r\sin\theta$. $\frac{x^3}{x^2+y^2} = \frac{r^3\cos^3\theta}{r^2} = r\cos^3\theta$. As $r \to 0$, $r\cos^3\theta \to 0$ regardless of $\theta$. Limit = 0. Q8: Continuity of rational function

Where is $f(x,y) = \frac{x+y}{x-y}$ continuous?

Solution: $f$ is a rational function, continuous everywhere except where denominator $=0$, i.e., $x-y=0$ or $y=x$. It's continuous on $\mathbb{R}^2 \setminus \{y=x\}$. Q9: Limit of composition

$\lim_{(x,y)\to(0,0)} \sin\left(\frac{x^2+y^2}{x^2+2y^2}\right)$

Solution: $\frac{x^2+y^2}{x^2+2y^2}$ is bounded between $\frac{1}{2}$ and $1$ (when $y=0$, it's $1$; when $x=0$, it's $1/2$). But the limit as $(x,y)\to(0,0)$ depends on path. Along $y=0$: ratio $=1$, $\sin(1)$. Along $x=0$: ratio $=1/2$, $\sin(1/2)$. Different limits ⇒ limit of the composition DNE. Q10: Epsilon-delta proof

Prove $\lim_{(x,y)\to(0,0)} 3x - 2y + 1 = 1$ using $\epsilon$-$\delta$.

Proof: Given $\epsilon > 0$, choose $\delta = \epsilon/5$ (or $\epsilon/3$). If $\|(x,y)\| = \sqrt{x^2+y^2} < \delta$, then $|3x-2y+1-1| = |3x-2y| \leq 3|x| + 2|y| \leq 5\sqrt{x^2+y^2} < 5\delta = \epsilon$. Thus $\lim f = 1$. ✓

🔗 Cross-References

• Next topic: Partial Derivatives (review)
• Week 10 (Critical Points): Continuity is needed for the extreme value theorem
• Week 11 (Differentiability): Differentiability implies continuity
• BSMA1001 (Maths 1): Single-variable limits and continuity Join Discord Previous9.2 Gradient & Directional DerivativesNext10.1 Tangent Planes