Limits (Algebraically)

Limits Infinities and Zeros

It is useful to have the following symbolic fractions when dealing with limits.  Note that infinity means positive or negative infinity.

1.      infinity
=   infinity
finite

2.               0
=   0
finite nonzero

3.      finite
=  0
infinity

4.     finite nonzero
=   infinity
0

5.      infinity
=  Do Algebra
infinity

6.     0
=   Do Algebra
0

Example:

1. 2. 3. The algebra that we can do is factoring.  We factor to get 4. We rationalize the denominator by multiplying by the conjugate root: Calculate: Exercises

Evaluate the following limits or state that they do not exist.

1. 2. 3. 4. Limits and Trigonometry

sin x
y  =
x

and discover that Corollary: Proof of the corrolary: Bye the first theorem, the first fraction approaches 1 as x approaches 0.  The second fraction evaluates to zero, hence the total expression is 0.

Applications:

1. 2. We set

u = 2x

then as x goes to zero so does u.  we have

x = u/2

substituting we get Exercise

Find The Squeeze Theorem

The squeeze theorem says that if a function  f is between two functions that have the same limit, then f has that limit also.

 The Squeeze Theorem Let            h < f < g   in an open interval containing c, except possibly at c.  If then Application

Show that Proof:

We have

-x  <  xsin(1/x)  x    for all   x

Since the squeeze theorem tells us that Another example of the squeeze theorem is here.