Indeterminate Forms of Limits by DerekFine

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									                                 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

								
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