chapter 2: limits 2.2 the limit of a function. limits the limit of f(x), as x approaches a, equals l...
DESCRIPTION
Helpful notes… In limits, x ≠ a This means we never consider that x = a The only thing that matters is how f(x) behaves near aTRANSCRIPT
Chapter 2: Limits
2.2The Limit of a Function
Limits
• “the limit of f(x), as x approaches a, equals L”• If we can make the values of f(x) arbitrarily
close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a
Lxfax
)(lim
Helpful notes…
• In limits, x ≠ a• This means we never consider that x = a
• The only thing that matters is how f(x) behaves near a
Example 1
• Guess the value of 11lim 21
xx
x
Example 2
• Estimate the value of 2
2
0
39limt
tt
Example 3
• Guess the value of xx
x
sinlim0
Example 4
• Investigate xx
sinlim0
Example 5
• Find
000,10
5coslim 3
0
xxx
Example 6
• The Heaviside function H is defined by
• What is the limit?
0,10,0
)(tt
tH
One sided limits
• Left-hand limit of f(x) as x approaches a
• Approaches from the negative side
Lxfax
)(lim
One sided limits
• Right-hand limit of f(x) as x approaches a
• Approaches from the positive side
Lxfax
)(lim
Therefore…
Lxfax
)(lim
If and only if…
andLxfax
)(lim Lxfax
)(lim
Example 7• The graph of a function g is shown in Figure 10
on page 71. Use it to state the values (if they exist) of the following:
)(lim
)(lim
)(lim
2
2
2
xg
xg
xg
x
x
x
Example 7• The graph of a function g is shown in Figure 10
on page 71. Use it to state the values (if they exist) of the following:
)(lim
)(lim
)(lim
5
5
5
xg
xg
xg
x
x
x
Example 8
• Find if it exists20
1limxx
Definition
• Let f be a function defined on both sides of a, except possibly at a itself
• Then
• Means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a, but not equal to a
• Happens in cases of functions with asymptotes!
)(lim xfax
Definition
• Let f be a function defined on both sides of a, except possibly at a itself
• Then
• Means that the values of f(x) can be made arbitrarily large negative by taking x sufficiently close to a, but not equal to a
• Happens in cases of functions with asymptotes!
)(lim xfax
Vertical Asymptotes
• The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:
)(lim
)(lim
)(lim
xf
xf
xf
ax
ax
ax
Example 9a
• Find 3
2lim3 x
xx
Example 9b
• Find 3
2lim3 x
xx
Example 10• Find the vertical asymptotes of f(x) = tan x
Homework
• P.74
• 4 – 9, 21, 25, 29