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Applications of Derivatives · Applied Problems

L'Hopital's Rule (Applications)

lim⁡x→af(x)g(x)=00,∞∞lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} \overset{\frac{0}{0}, \frac{\infty}{\infty}}{=} \lim_{x \to a} \frac{f'(x)}{g'(x)}

L'Hopital's Rule applied to evaluate limits involving indeterminate forms such as 0·∞, ∞-∞, 0⁰, ∞⁰, 1^∞ by algebraic rearrangement to 0/0 or ∞/∞.

Worked examples

Find lim(x→0⁺) x ln x (form 0·(-∞)).
  1. Rewrite as lim(x→0⁺) ln x / (1/x) -form -∞/∞
  2. L'Hopital: lim (1/x) / (-1/x²) = lim (-x) = 0

Answer: 0

Find lim(x→0⁺) xˣ (form 0⁰).
  1. Let y = xˣ, so ln y = x ln x
  2. From previous example, lim(x→0⁺) x ln x = 0
  3. So lim ln y = 0, thus lim y = e⁰ = 1

Answer: 1

Related formulas

L'Hopital's Rule

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