Calculus: Analytic Geometry and Calculus, with Vectors

3.2 Limits 131

9 Supposing that y = x2 and y +i y = (x +Ax)2, show that

10

lim Ly = 2x.

,iz-.0 Ox

Prove that

lim (x + x3= 3x2.

11

AX-0 Ax

Prove that, when x > 0,

lim V(x + Ax)3 - 1x3=2tix. 3

.%x-O ,x

12 We have shown that

sin (x+h) -sinx

h

sihhcosx-l - hoshsinx

and we shall learn that

limsin h= 1

h-.o At

lim1-cosh=0

h-.o h

Use these facts to find that

limsin (x + h) - sin x= cos X.

h-0 h

13 Supposing that y 54 0, prove that

z z

lim

z--.0x2+y2

14 Supposing that y = 0, prove that

xz-yz

lim--=1.

X-Oxz+y2

(^15) Prove that if lim f(x) = L, then to each positive number a there corre-
sponds a positive number 3 such that
jf(x2) - f(xi)I < E
whenever 0 < Ix2 - al < 5 and 0 < Ix1 - al < 5. Remark: Proof of this result
depends upon the idea that it two things are near the same place, then the things
must be near each other. The details require careful attention, however. To
prove the result, let e be a positive number. Then e/2 is a positive number.
Hence there is a positive number S such that If(x) - LI < E/2 whenever 0 <
Ix - al < S. Therefore,
l f(xs) - f(xl) I = I U(x2) - L] - U(xi) - LI!
5lf(x2)-LI+lf(xi)-LI<2+2=E
whenever 0 < Ix1 - a! < 8 and 0 < (x2 - al < 3.