4.2 Limits and Continuity

← Chapter 4 §4.2 Limits and Continuity 极限与连续 · Chapter 4 · Differentiation of Functions of Several Variables

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condition	here
1. $f(0,0)$ exists
2. the limit exists
3. they are equal

Objective	Takeaways; say them first, then click
4.2.1 Calculate a limit in two variables.	δ disk; every ε band has a δ disk; check the bottom, then substitute
4.2.2 Paths and boundary points.	two paths, two limits ⇒ DNE; all lines agreeing proves nothing; at the edge, approach through the domain
4.2.3 Conditions for continuity.	value exists; limit exists; equal
4.2.4 Verify continuity at a point.	check all three (Example 4.11); sums, products, compositions of continuous pieces
4.2.5 Three or more variables.	δ ball; same laws: Example 4.13 → 25/16

Mistake	Result	Fix
Two paths agree, so the limit exists	Example 4.9b: every line gives 0, still no limit	agreement proves nothing; prove with a bound
Trying only lines $y=kx$	missing $x=y^2$ , where Example 4.9b is 1	try curves too, or polar coordinates
"0/0, so no limit"	$\sin(x^2+y^2)/(x^2+y^2)\to1$	0/0 means work to do
Quotient law with a bottom tending to 0	a wrong or meaningless value	check the bottom first: 11 in Example 4.8b
Continuity from the limit alone	$x^2y/(x^2+y^2)$ has limit 0 but no value at the origin	all three conditions
Polar limit that still depends on θ	$2\cos\theta\sin\theta/(3\cos^2\theta+\sin^2\theta)$ called "the limit"	a θ left over means no limit

#	task	answer
4.6	$\lim\sqrt[3]{\dfrac{x^2-y}{y^2+x-1}}$ at $(5,-2)$	$\sqrt[3]{27/8}=\tfrac32$
4.7	$\dfrac{(x-2)(y-1)}{(x-2)^2+(y-1)^2}$ at $(2,1)$	along $y-1=k(x-2)$ : $\tfrac{k}{1+k^2}$ , so no limit
4.8	$\sqrt{29-x^2-y^2}$ at $(5,-2)$	$0$ (a boundary point)
4.9	$\sqrt{26-2x^2-y^2}$ continuous at $(2,-3)$	$f(2,-3)=\sqrt{9}=3=$ limit (the key prints 16 for 26)
4.10	$2x^2y^3+3$ and $(2x^2y^3+3)^4$	products, sums, a composition: continuous everywhere
4.11	$\sqrt{13-x^2-2y^2+z^2}$ at $(4,-1,3)$	$\sqrt4=2$

#	task	answer
61	$\dfrac{5x^2y}{x^2+y^2}$ at $(1,2)$	$2$
63	$\dfrac{4x^2+10y^2+4}{4x^2-10y^2+6}$ at $(0,0)$	$\tfrac23$
65	$\dfrac{y^2\sin x}{x}$ at $(0,1)$	$1$
67	$\dfrac{y\tan x}{y+1}$ at $(\pi/4,1)$	$\tfrac12$
69	$\tfrac1x-\tfrac5y$ at $(2,5)$	$-\tfrac12$
71	$e^{-x^2-y^2}$ at $(4,4)$	$e^{-32}$
73	$x^2y^3-x^3y^2+3x+2y$ at $(1,2)$	$11$
75	$\dfrac{xy+1}{x^2+y^2+1}$ at $(0,0)$	$1$
77	$\ln(x^2+y^2)$ at $(0,0)$	does not exist: $\to-\infty$
79	a boundary point of $R$ is …	one whose every open disc holds points inside and outside $R$
81	$\dfrac{x^4-4y^4}{x^2+2y^2}$ at $(0,0)$	$0$ : it equals $x^2-2y^2$
83	$\dfrac{x^2-xy}{\sqrt x-\sqrt y}$ at $(0,0)$	$0$ : it equals $x(\sqrt x+\sqrt y)$
85	$\dfrac{x^2-y^2-z^2}{x^2+y^2-z^2}$ at the origin	does not exist: $x$ -axis 1, $y$ -axis $-1$
87	$\dfrac{xy+y^3}{x^2+y^2}$ at $(0,0)$	does not exist: axes 0, $y=2x$ gives $\tfrac25$
89	$\dfrac{x^2y}{x^4+y^2}$ at $(0,0)$	does not exist: axes 0, $y=x^2$ gives $\tfrac12$
91	$\ln(x+y)$	continuous where $y>-x$
93	$1/(xy)$	continuous where $x\ne0$ and $y\ne0$
95	$\dfrac{x^2y}{x^2+y^2}$ , with $f(0,0)=0$	continuous at $(0,0)$ : $\|f\|\le\|y\|$
97	$\dfrac{x^2-y^2}{x^2+y^2}$ at $(0,0)$	discontinuous: no limit, no value
99	$\arctan\dfrac{xy^2}{x+y}$	continuous except on $y=-x$
101	$x^2+y^2-2z^2$	continuous at every point of space
103	$\dfrac1{x^2+y^2}$ at $(0,0)$	no limit: grows without bound (continuity lab)
107	$\dfrac{\sin\sqrt{x^2+y^2}}{\sqrt{x^2+y^2}}$ at $(0,0)$	$1$ : in polar form $\sin r/r$
109	$f(g(x,y))$ , $f(t)=1/t$ , $g=2x-5y$	continuous off the line $2x-5y=0$
111	$\lim_{h\to0}\frac{f(1+h,y)-f(1,y)}{h}$ , $f=x^2-4y$	$2$

Limits and Continuity

In the plane, “close to $(a,b)$ ” means inside a small δ disk

$\lim f=L$ : every ε band around $L$ has a δ disk whose values stay inside it

Some δ disk must fit inside every ε band

Limits pass through sums, products, quotients, powers and roots

Check the bottom’s limit first; if it is not 0, substitute

Each law is the one-variable proof with distance in the plane

In the plane there are infinitely many ways in — the limit must agree on all

$\dfrac{2xy}{3x^2+y^2}$ is 0 along $y=0$ but $\tfrac12$ along $y=x$ : no limit at the origin

Along $y=kx$ the value is $\dfrac{2k}{3+k^2}$ : each direction has its own height

Polar coordinates: a bound in $r$ alone proves a limit, a leftover θ disproves it

Every line gives 0, yet the parabola $x=y^2$ gives 1: still no limit

Compare the heights along lines and parabolas

Along lines through $(2,1)$ the value is $\dfrac{k}{1+k^2}$ : no limit

Every line near the origin slides onto flat parabolas, so it misses the ridge

At a boundary point of the domain, approach only through the domain

A point is interior if some disk fits inside $S$ , boundary if every disk pokes out

Open leaves its edge out, closed keeps all of it — half an edge is neither

$\sqrt{29-x^2-y^2}$ at the rim point $(5,-2)$ tends to 0

Continuous at $(a,b)$ : the value exists, the limit exists, and they are equal

Verify continuity by checking all three conditions at the point

Four functions at the origin: which condition fails?

Sums, products and compositions of continuous functions are continuous

Name the continuous pieces and the theorem that joins them

In ε–δ form, continuity lets $L$ be $f(a,b)$ — and sums stay continuous

Composition: chain the two ε–δ promises

In three variables the δ disk becomes a δ ball — the limit laws are unchanged

Three variables, same routine: bottom first, then top, then divide

Shrink the δ ball and $f$ closes in on $\tfrac{25}{16}$

A limit in the plane must agree along every path; continuity is limit = value

Common mistakes

Practice and answers