Limit

Limit

1. What Limit Means for $f(x,y)$

For $f(x,y)$ near $(a,b)$, we ask: does $f(x,y)$ approach a single number $L$ no matter how $(x,y)→(a,b)$?

means: for every $ε>0$, there is $δ>0$ such that if $0<√[(x-a)²+(y-b)²]<δ$ then $|f(x,y)-L|<ε$. Intuitively: values get arbitrarily close to $L$ as point gets arbitrarily close to $(a,b)$ from any direction.

Key difference from one variable: path matters. In $y=f(x)$, approach is left/right only; in two variables, infinitely many paths (lines, parabolas, spirals) must all give same $L$, or limit fails.

2. Two-Path Test — The Fast Way to Show Non-Existence

If you find two paths approaching $(a,b)$ giving different limiting values, the joint limit does not exist.

Paths to try first: y = m x  (different slopes m), y = k x² , x =0, y=0

If both give same $L$, you have not proved existence — need general squeeze or ε-δ — but if they differ, you are done.

3. Working by Conversion

Often set $x = a + r cosθ, y = b + r sinθ$ and let $r→0$. If limit depends on $θ$, no limit; if $|f -L| ≤ k r^α →0$ uniformly in $θ$, limit is $L$ (squeeze).

4. Worked Examples

(a) f(x,y)= x y / (x² + y²) at (0,0):
    Along y = m x : f = m x²/(x²+m²x²)= m/(1+m²) depends on m
    m=0 ⇒ 0 ; m=1 ⇒ 1/2 ⇒ different ⇒ limit DOES NOT EXIST          ✓
    Along y=0 →0 ; along y=x →1/2 is enough.

(b) f(x,y)= x² y / (x² + y²) at (0,0):
    Convert: |f| = |x² y|/(x²+y²) ≤ |y| (since x²/(x²+y²)≤1) ; as (x,y)→0, |y|→0
    So f→0 by squeeze, although two-path gave 0 both ways. General proof:
    r,θ: f = r cos²θ · r sinθ = r·cos²θ sinθ →0 uniformly ⇒ LIMIT 0      ✓

(c) f(x,y)= (x⁴ - y⁴)/(x² + y²) at (0,0):
    Use polar: x=r cosθ, y=r sinθ ⇒ f= r²(cos⁴ - sin⁴)= r² cos2θ →0 ⇒ LIMIT 0 ✓

(d) f(x,y)= (x² y²)/(x² + y²)^{3/2}?? Need check: use r: = r cos² sin² →0 ✓

(e) f(x,y)= (y - x²)(y -2x²)/[x⁴+(y - x²)²] tricky: along y=k x² gives different limits
    ⇒ does not exist — shows why parabola paths matter beyond lines.

(f) lim (x,y)→(1,2) (x+y)/(x²+y²) : continuous there (denominator ≠0) ⇒ 3/5 directly ✓

5. Limit Laws

If $lim f = L, lim g = M$ (both exist), then $lim (f±g)=L±M$, $lim fg = LM$, $lim f/g = L/M$ (if $M≠0$). Composition with continuous outer function preserves limit.

6. Relation to Single-Variable

Iterated limits $lim_{x→a} lim_{y→b} f$ and $lim_{y→b} lim_{x→a} f$ may each exist but differ, or differ from joint limit. Existence of joint limit implies both iterated limits equal it (if they exist), but converse false — so never deduce joint limit from iterated.

7. Exam Points

• Show $lim_{(x,y)→(0,0)} xy/(x²+y²)$ or $x²y/(x²+y²)$ does not exist / exists — 5 marks, very frequent.
• Two-path test with $y=mx$ and $y=kx²$ — 3-5 marks.
• Polar / squeeze proof for limits that do exist — 5 marks, frequent.
• Distinguish joint vs iterated vs path limits — 3 marks.