Question

Noah

想問第4題畫線處

分母(tan²x)-1
=(sin²x/cos²x)-1
=(sin²x-cos²x)/cos²x

為什麼把1/cos²x移到分子以後
分母不是sin²x-cos²x ?
感謝！

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
●

微積分 極限

Answer 1

✨ 最佳解答 ✨

我就是我

應該是寫錯
sec²x-1=tan²x 是恆等式
1/tan²x確實才會等於cos²x/sin²x

tan²x-1應改為tan²x

Noah

感謝！
我知道sec²x-1=tan²x 是恆等式，
可是我在想，當x趨近於0，sin²x≈0，sec²x≈tan²x ？

我就是我

tan²x-1沒辦法變cos²x/sin²x
而且這裡不是在算極限，只是純粹的等式
化簡之後才放回極限裡運算

Noah

學到了，感謝❤