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CHAPTER 41
41.48
41.49
41.50
41.51
41.52
41.53
What is signified by
Recall that |(M, i>)| ••
Then
f(x, y) = L means that, for every e > 0, there
whenever \(x, y) — (a, b)\ < 8.
such that \f(x, y)- L\ exists 8 > 0
Prove that
2x - 3y = -4.
Let e>0. We must find 5>0 such that \(x, y) - (1,2)1 <8 implies \(2x -3y) - (-4)1 < e; that is,
<5 implies |2x-3y+4| Assume
Then |*-1|<5 and |y-2|<5. So, \2x - 3y + 4| = \2(x - 1) - 3(y -2)| s2|jc - 1| + 3\y - 2\ <28 +
3S=5S Find
if it exists.
Note that, if y = mx and (jc, y)-*(0,0), then
However, this
is not enough to ensure that 2xy l(x + y )-»0 no matter how (x, y)-»(0,0). Take any e>0. Note
and y
2 =£ x
2 + y
2 . So,
Observe also that
that K*, y) - (0,0)| =
Thus,
Hence, if we choose S = e/2 and if
then
if it exists.
Find
Let v = mx. Then
as (x, y)—*(0,0). However, if we let x = y
2 , then
Hence, as (x, y)->(0,0)
along the parabola x = y
2 ,
Therefore,
does not exist.
Find
if it exists.
Let y = mx. Then
Since (1 — m2)/(l + m2) depends on m, (x2 — y2)/(x2 +y2) approaches different numbers as (x, y)—»(0,0)
does not exist.
along different lines. Hence.
Find
if it exists.