Nested Indeterminate Forms

Suppose ff and gg are differentiable functions such that

  1. g(x)0g'(x) \neq 0 on an open interval II containing 00;
  2. limx0f(x)=0\lim_{x \to 0} f(x) = 0 and limx0g(x)=0\lim_{x \to 0} g(x) = 0;
  3. limx0f(x)g(x)=0\lim_{x \to 0} \frac{f'(x)}{g'(x)} = 0.

L'Hôpital's Rule concludes that we can find limx0f(x)g(x)=limx0f(x)g(x)=0\lim_{x \to 0} \frac{f(x)}{g(x)} = \lim_{x \to 0} \frac{f'(x)}{g'(x)} = 0.

What can we conclude about how to find limx0f(f(x)g(x))g(f(x)g(x))\lim_{x \to 0} \frac{f(\frac{f(x)}{g(x)})}{g(\frac{f(x)}{g(x)})} ?

Example question: put f(x)=sin(x)xf(x) = \sin(x) - x and g(x)=xg(x) = x, find limx0sin(sin(x)xx)(sin(x)xx)(sin(x)xx)=limx0sin(sin(x)x1)(sin(x)x1)(sin(x)x1)\lim_{x \to 0} \frac{\sin(\frac{\sin(x) - x}{x}) - (\frac{\sin(x) - x}{x})}{(\frac{\sin(x) - x}{x})} = \lim_{x \to 0} \frac{\sin(\frac{\sin(x)}{x} - 1) - (\frac{\sin(x)}{x} - 1)}{(\frac{\sin(x)}{x} - 1)} ?

Note by A Former Brilliant Member
2 weeks, 4 days ago

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@Aruna Yumlembam

Yajat Shamji - 2 weeks, 4 days ago

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@Zakir Husain

Yajat Shamji - 2 weeks, 4 days ago

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@Neeraj Anand Badgujar

Yajat Shamji - 2 weeks, 4 days ago

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@Yajat Shamji Do you want the solution of that last limit problem?

Lil Doug - 2 weeks, 4 days ago

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Yes! I think the Calculus Geeks can handle this!

If you don't know where Calculus Geeks came from, ask Aruna Yumlembam...

Yajat Shamji - 2 weeks, 3 days ago

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@Naren Bhandari

Yajat Shamji - 2 weeks, 4 days ago

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