On-line Math 21

1.3.1  Examples of computing limits using rules and theorems.

Example 1 Compute

lim x® 3  x2+2x-1 x3-3x2+2 .

Example 2 Compute

lim x® -1  Ö   x3+2x+7   .

To be fair, we really don't have the theory to claim this (yet), but we can argue simply that, since

lim x® -1  x3+2x+7 = lim x® -1  æ è Ö   x3+2x+7   ö ø 2   = lim x® -1  Ö   x3+2x+7   lim x® -1  Ö   x3+2x+7   ,

and

lim x® -1  x3+2x+7 = (-1)3+2(-1)+7 = 4,

then

lim x® -1  Ö   x3+2x+7   = 2.

Example 3 Compute

lim x® 9  x2-81 Öx-3 .

Example 4 Define

f(x): = ì ï í ï î   ___ Öx-4   , if x > 4 8-2x, if x < 4 .

Find

lim x® 4  f(x),

if it exists.

Example 5 Find

lim x® 0  xsin( 1 x )

by using the squeeze theorem.

By David L. Johnson, last modified 3/10/00.