# IBDP Past Year Exam Questions – Limits

1. [N18/P3/TZ0]

(a) Use L’Hôpital’s rule to determine the value of  . [5 marks]

(b) Hence find  .   [3 marks]

2. [M17/P3/TZ0]

Use l’Hôpital’s rule to determine the value of  $\underset{x\to 0}{lim}\frac{{\mathrm{sin}}^{2}x}{x\mathrm{ln}\left(1+x\right)}$ .  [7 marks]

3. [N16/P3/TZ0]

Using l’Hôpital’s rule, find  $\underset{x\to \infty }{lim}\left(\frac{arc\mathrm{sin}\left(\frac{1}{\sqrt{x+1}}\right)}{\frac{1}{\sqrt{x}}}\right)$ .  [6 marks]

4. [M16/P3/TZ0]

Determine the exact value of  .   [3 marks]

5. [M16/P3/TZ0]

Use l’Hôpital’s rule to find  $\underset{x\to \infty }{lim}{x}^{2}{e}^{–x}$ .  [4 marks]

6. [M17]

Using l’Hôpital’s rule, show that   where  $n\in \mathrm{ℕ}$ .  [4 marks]

7. [M16]

Use l’Hôpital’s rule to show that  $\underset{x\to \infty }{lim}\frac{{x}^{3}}{{e}^{x}}=0$ .   [3 marks]

8. [M14]

Using l’Hôpital’s rule, show that  .

9. [M11/P3/TZ0]

(a) Find the first three terms of the Maclaurin series for $\mathrm{ln}\left(1+{e}^{x}\right)$ . [6 marks]
(b) Hence, or otherwise, determine the value of  . [4 marks]

10. [N11/P3/TZ0]

Find

11. [M09/P3/TZ0]

Find

(i)    ;    [4 marks]

(ii)    .    [7 marks]

12. [N10/P3/TZ0]

Find  $\underset{x\to 0}{lim}\left(\frac{1–\mathrm{cos}{x}^{6}}{{x}^{12}}\right)$ .    [7 marks]