[ Day 2 ]
3.5 Limits at Infinity
Limits Involving Trigonometric Functions Find:
sin x lim x
A.
x
As , the sine function oscillates between 1 and ­1,
...
sin x does
lim x
NOT
exist!
sin x lim
x x
B.
­1 sin x 1
since
, x > 0
­1 sin x 1
x
x
x
where,
­1
lim
x = 0
x
and 1
lim
x = 0
x
Squeeze Thm. (1.3),
... by the sin x lim
x = 0
x
1
examples:
Numerical and Graphical Analysis. [p199 #10] . . . complete the table, and then use the graph to estimate the limit graphically. f (x) = 8x
x2 ­ 3
numerically
100 101 102
x
106
f (x)
graphically
2
[p199 #26] Find the limit (analytically). x
lim
­
x + 1
x
2
3
Infinite Limits at Infinity
useful in analyzing the "end behavior" of its graph
apply to (3.6) curve sketching
examples:
Find each limit.
1.
lim x4
x
2x ­ 4x 2. xlim
­ x + 1 2
Recall:
For rational functions (having no common factors) ­
if the degree of the numerator is one greater than the degree of the denominator, then an oblique (slant) asymptote will be yielded ­ after long division! 4
Concepts to Review!
useful in analyzing the graph of f (x)
Intercepts (section P.1) p.4
x­intercept: (x, 0)
y­intercept: (0, y)
Symmetry (section P.1) p.5
symmetric to y­axis
(­x, y) (x, y)
symmetric to x­axis
(x, ­y) (x, y)
. .
.
.
.
symmetric to origin
(­x, ­y) (x, y)
Even
.
Odd
Domain & Range (section P.3) p.3
Continuity (section 1.4)
Vertical Asymptotes (section 1.5)
Differentiability (section 2.1)
5
example:
[p200 #50] Sketch the graph. Look for extrema, intercepts, symmetry, and asymptotes. Use a graphing utility to verify.
y =
x ­ 3
x ­ 2
6
7
Assignment:
p200 #45­48, 49, 51, 55, 57 8