1. LIMITS

2. We say that the limit of f(x), as x approaches a written
lim f (x) = L
x a
if we can take outputs f(x) as close to L as we wish by taking
inputs x sufficiently close to a but not equal to a.

3. x2 ­ 9
Graph the function f(x) =
x ­ 3

4. Now lets estimate lim f(x)
x 3
On graphing calculator press 2nd [Tblset]

5. Then press 2nd [TABLE]

6. Graph f(x) = sinx
x

7. estimate lim f(x)
x 0

10. Graph f(x) = x
x

12. x lim x
lim = ­1
x 0+
x = 1 x 0­ x
Right ­ hand limit: lim f(x) = L means that f(x) can be
x a +
made as close to L as we wish by taking x sufficiently close
to a but greater than a
Left ­ hand limit: lim f(x) = L means that f(x) can be
x a­
made as close to L as we wish by taking x sufficiently close
to a but less than a

13. lim sinx = lim sinx
+ = lim sinx = 1
x 0
x x 0 x ­ x 0
x
lim x
lim
= x 0
x lim x
therefore, x 0 does not exist
x 0+
x ­
x x

14. Rules for calculation limits
1. Sum Rule: lim [ f(x) + g(x) ] = lim f(x) + lim g(x) = L + M
x a x a x a
2. Difference Rule: lim [ f(x) ­ g(x) ] = lim f(x) ­ lim g(x)
x a x a x a
= L ­ M
x a
. .
3. Product Rule: lim [ f(x) g(x) ] = lim f(x) lim g(x) = LM
x a x a

15. 4. Constant Multiple Rule: lim k f(x) = kL
x a
f(x) lim f(x)
lim
x a L =
if M 0
5. Quotient Rule: x a = =
g(x) lim g(x) M
x a

16. Find lim x2
x 2
3x + 4
lim x2
lim x2 x 2
=
x 2
3x + 4 lim (3x + 4)
x 2
lim x lim x
= x 2 x 2
3 lim x + lim 4
x 2 x 2
2
5

17. 2
Find lim x ­ 5x + 6
x 2
x ­ 2

18. Questions 1 ­ 19 odd only
Exercise 2.7