Limit - Ans.fyi 參考答案

Limit

知識就是力量

2010-07-01 4:44 am

Without using differentiation, find lim_h->0 [f(x+h) - f(x)]/h when f(x) = 3√x

更新1:

3√x = x^1/3

更新2:

beautiful

回答 (1)

六呎將軍

2010-07-01 5:42 am

✔ 最佳答案

lim (h → 0) [f(x + h) - f(x)]/h

= lim (h → 0) [3√(x + h) - 3√x]/h

= lim (h → 0) [3√(x + h) - 3√x][3√(x + h)2 + 3√(x + h)3√x + 3√x2]/{h[3√(x + h)2 + 3√(x + h)3√x + 3√x2]}

= lim (h → 0) [3√(x + h) - 3√x][3√(x + h)2 + 3√(x + h)3√x + 3√x2]/{h[3√(x + h)2 + 3√(x + h)3√x + 3√x2]}

= lim (h → 0) h/{h[3√(x + h)2 + 3√(x + h)3√x + 3√x2]}

= lim (h → 0) 1/[3√(x + h)2 + 3√(x + h)3√x + 3√x2]

Directly sub h = 0, we have:

lim (h → 0) 1/[3√(x + h)2 + 3√(x + h)3√x + 3√x2]

= 1/[3√x2 + 3√x3√x + 3√x2]

= 1/(33√x2)

參考: Myself

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