• Title: Other L’Hospital’s Rules

  • Series: Real Analysis

  • Chapter: Differentiable Functions

  • YouTube-Title: Real Analysis 43 | Other L’Hospital’s Rules

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  • Quiz Content

    Q1: Let $f,g \colon \mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that $\lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)}$ makes sense and exists. What is a correct implication using l’Hospital’s rule?

    A1: If $\displaystyle \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x)$, then $ \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)} $.

    A2: If $\displaystyle \lim_{x \rightarrow \infty} f(x) \neq \lim_{x \rightarrow \infty} g(x)$, then $ \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)} $.

    A3: If $\displaystyle \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = 0$, then $ \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)} $.

    A4: If $\displaystyle \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = 1$, then $ \displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \infty} \frac{f^\prime(x)}{g^\prime(x)} $.

    Q2: Let’s apply l’Hospital’s rule for the following limit for $a > 0$: $$ \lim_{x \rightarrow \infty} \frac{ \log(1 + a x) }{ x } $$

    A1: $\frac{1}{a}$

    A2: $1$

    A3: $0$

    A4: $a$

    A5: $2a$

    Q3: Let’s apply l’Hospital’s rule for the following limit: $$ \lim_{x \rightarrow 0} x \log(x) $$

    A1: $\frac{1}{2}$

    A2: $1$

    A3: $0$

    A4: $-1$

    A5: $\infty$

    Q4: Let’s apply l’Hospital’s rule (several times) for the following limit: $$ \lim_{x \rightarrow \infty} \frac{x^5}{\exp(x)} $$

    A1: $\frac{1}{2}$

    A2: $1$

    A3: $0$

    A4: $-1$

    A5: $\infty$

  • Last update: 2025-01

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