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: $\infty$