On-line Math 21

1.4  Computing limits

How do you compute limits? There are a number of tricks to computing limits, which we'll get to more fully. But, for now, note that we are using some obvious facts:

Fact 1 If you know the limits of f(x) and g(x) at x=a, then:

The limit of a sum is the sum of the limits:

lim x® a  (f(x)+g(x))= lim x® a  f(x)+ lim x® a  g(x),

The limit of a product is the product of the limits

lim x® a  (f(x)g(x))= lim x® a  f(x) lim x® a  g(x),

and the limit of the quotient is the quotient of the limits,

lim x® a  f(x) g(x) = lim x® a  f(x) lim x® a  g(x) .

This last one only makes sense if

lim x® a  g(x) ¹ 0.

If it is zero, we have more work to do to find that limit.

At the basis of the computations we did earlier, though, we really should add a few more facts, which seem kind of obvious, but really form the basis of the computations with limits:

Fact 2 Obvious limits:

lim x® a  x=a.

lim x® a  1=1.

If f(x)=g(x) for all x near a, except perhaps at a itself, then

lim x® a  f(x)= lim x® a  g(x).

The first two in this list are pretty obvious indeed, but the third one really is the way you first work with limits. let's use it to do the first limit we did,

lim x® 1  x2-1 x-1 .

If

f(x)= x2-1 x-1 ,

then for all x ¹ 1, f(x)=x+1, since

x2-1 x-1 = (x+1)(x-1) x-1 = x+1, by cancelling the (x-1)¢s,

so the limit

lim x® 1  x2-1 x-1 = lim x® 1  x+1=2.

That last part really uses the first two stupid limits and the addition rule, but that is getting awfully picky.

1.5  A bit more theory:

These two results may also seem rather obvious, but they will help make sense of some rather nasty limits.

Theorem 3 If f(x) £ g(x) for all x in an open interval that contains a (except possibly at a itself, then

lim x® a  f(x) £ lim x® a  g(x).

Theorem 4 If f(x) £ g(x) £ h(x) in an open interval that contains a (except possibly at a itself, and

lim x® a  f(x)=L= lim x® a  h(x),

then

lim x® a  g(x)=L.

This result is usually used to show results such as

Example 5

lim x® 0  x sin æ ç è 1 x ö ÷ ø =0.

Solution

Example 6 Compute

lim x® 3  x2-x-6 x3-4x2+3x .

Hints

Example 7 Compute

lim x® -1  Ö   x3+2x+7   .

Solution

Exercise 2 Compute

lim x® 3  x2-9 x2-2x-3 .

Exercise 3 Compute

lim x® 2  x2-5x+6 x2-4 =

Exercise 4 Compute

lim x® 9  x2-81 Öx-3 =

Exercise 5 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.