# Indeterminate Forms of Limits by DerekFine

VIEWS: 865 PAGES: 1

• pg 1
```									                                 Indeterminate Forms of Limits

Form                                    Example 1                           Example 2

0                                      sin x                            x3 − 1
lim      =1                     lim        =3
0                                   x→0 x                           x→1 x − 1

∞                                    x2     1                                x3
lim        =                              lim     =0
∞                             x→∞ 1 + 3x2   3                            x→∞ e2x

1     1               1
∞−∞                         lim+              −              =        lim (sec x − tan x) = 0
x→1            ln x x − 1             2   x→ π +
2

0·∞                                           1
lim x sin x = 1                         lim x ln x = 0
x→∞                                  x→0+

00                                    lim+ xx = 1                  lim+ (e−1/x )x = e−1
x→0                          x→0

x
∞                                          1
1                             lim           1+          =e        lim (1 + sin x)cot x = 1
x→∞               x                 x→0+

∞0                             lim (tan x)cos x = 1                   lim (ex )1/x = e
x→ π −
2
x→∞

The following forms are determinate.
∞+∞→∞

−∞ − ∞ → −∞

(0+ )∞ → 0

(0+ )−∞ → ∞

0
→0
∞
∞           ∞
+
→ ∞ and    → −∞
0           0−
Gilles Cazelais. Typeset with LTEX on January 26, 2008.
A

```
To top