IBDP Past Year Exam Questions – Limits

1. [N18/P3/TZ0]

(a) Use L’Hôpital’s rule to determine the value of  $\underset{x\to 0}{lim}\left(\frac{e^{–3x^2}+3\cos \left(2x\right)– 4}{3x^2}\right)$ . [5 marks]

(b) Hence find  $\underset{x\to 0}{lim}\left(\frac{{\int }_0^x\left(e^{–3t^2}+3\cos \left(2t\right)– 4\right)dt}{{\int }_0^{x}3t^{2}dt}\right)$ .   [3 marks]

2. [M17/P3/TZ0]

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

3. [N16/P3/TZ0]

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

4. [M16/P3/TZ0]

Determine the exact value of  $\underset{x\to 0}{lim}\frac{e^x\sin x– x– x^3}{x^3}$ .   [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   $\underset{x\to \infty }{lim}{x}^n{e}^{– x}=0$ where  $n\in \mathrm{\mathbb{N}}$ .  [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  $\underset{x\to \infty }{lim}\frac{x^n}{e^{\lambda x}}=0; n\in {\mathrm{\mathbb{Z}}}^{+} , \lambda \in {\mathrm{ℝ}}^{+}$ .

9. [M11/P3/TZ0]

(a) Find the first three terms of the Maclaurin series for $\ln \left(1+e^x\right)$ . [6 marks]
(b) Hence, or otherwise, determine the value of $\underset{x\to 0}{lim}\frac{2\ln \left(1+e^x\right)– x– \ln 4}{x^2}$  . [4 marks]

10. [N11/P3/TZ0]

Find  $\underset{x\to \frac{1}{2}}{lim}\left(\frac{\left({\displaystyle \frac{1}{4}– x^2}\right)}{cot \mathrm{\pi x}}\right)$

11. [M09/P3/TZ0]

Find

(i)    $\underset{x\to 0}{lim}\frac{\tan x}{x + x^2}$ ;    [4 marks]

(ii)    $\underset{x\to 1}{lim}\frac{1 – x^2 + 2x^2\ln x}{1 – \sin {\displaystyle \frac{\mathrm{\pi x}}{2}}}$ .    [7 marks]  

12. [N10/P3/TZ0]

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