Online test

Derivatives – test 2: L’Hôpital’s rule

10 multiple-choice and short-answer problems on derivatives (Mathematics I). The solutions are shown after checking.

For multiple-choice questions choose one answer. For the others type the answer: a number, a decimal or a fraction such as 5/12. When you finish, click Check answers.

1. Find \(\lim\limits_{x\to0}\frac{\sin4x}{2x-\sin3x}\).Enter a number only

2. Find \(\lim\limits_{x\to0}\frac{x\sin x}{\cos x-1}\).Enter a number only

3. Find \(\lim\limits_{x\to2}\frac{\ln(x^2-3)}{x^2+3x-10}\).Enter a number only

4. Find \(\lim\limits_{x\to1}\left(\frac1{x-1}-\frac1{\ln x}\right)\).Enter a number only

5. Find \(\lim\limits_{x\to0}\frac{x-\sin x}{x^3}\).Enter a number only

6. Find \(\lim\limits_{x\to0^+}x\ln x\).Enter a number only

7. What is \(\lim\limits_{x\to\infty}\left(1+\frac3x\right)^x\)?

A. \(3\)

B. \(e\)

C. \(1\)

D. \(e^3\)

8. What is \(\lim\limits_{x\to0}(\cos x)^{1/x^2}\)?

A. \(\sqrt e\)

B. \(1\)

C. \(\frac1{\sqrt e}\)

D. \(0\)

9. Find \(\lim\limits_{x\to0}\frac{e^x-e^{-x}-2x}{x-\sin x}\).Enter a number only

10. Find \(\lim\limits_{x\to0}\frac{x+1}{x+2}\).Enter a number only