x²
Example 4 Find lim
x 0 sec X 1
SOLUTION The evaluation of this limit requires a little imagination. Since both the
numerator and denominator tend to zero as x tends to zero, it is not clear what happens
to the fraction. However, we can rewrite the fraction in a more amenable form by
multiplying both numerator and denominator by sec x + 1.
x2
x²
sec x 1
lim
x-0 secx
1
-
=
=
x2
sec x 1
-
x² (secx + 1)
sec² x 1
lim (sin
x-0 sin x
sec x + 1
sec x + 1)
-
=
Since each of these factors has a limit as x tends to 0, the fraction we began with has a
limit:
=
x² (secx + 1)
tan² x - 1
x² cos²x(secx + 1)
sin² X
2
(*) ² (cos²x)(secx + 1).
2
·lim cos²x lim (secx + 1) = (1)(1)(2) = 2.
x→0
x
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