Infinite Limits

1
Infinite Limits

• We shall say that LimxgtcF(x) ? if
  F(x) can be made as large as we like by making x
  sufficiently close to c.
• In like manner, we define Limx-gtcF(x) -
  ?.

2
Vertical Asymptotes

• If F(x) approaches ? or - ? as x approaches c
  from the left or the right, we say that the line
  x c is a vertical asymptote of the graph of F(x)

3
Vertical Asymptotes

• Suppose F(x) and G(x) are continuous on an open
  interval containing c. If F(c) is not 0 and G(c)
  0, and there is an open interval about c so
  that G(x) ix not zero for x not equal to c, then
  the graph of H(x) F(x)/G(x) has a vertical
  asymptote at x c.

4
Infinite Limits

• Suppose that c and L are real numbers and F(x)
  and G(x) are functions such that Limx-gtcF(x) ?
  andLimx-gtcG(x)L.Then
• Limx-gtcF(x) ? G(x) ?.
• Limx-gtcF(x)G(x) ? if L gt 0 and -? if L lt0.
• Limx-gtc F(x)/G(x) 0.

5
CAUTION

• When we say that Limx-gtcF(x)
  ? ?, we are simply describing the behavior of the
  function in the vicinity of c. The limit does
  NOT exist at c.
• When we say that Limx-gtcF(x) exists, we mean that
  it is a real number.