A2 June 2024 Q3
3. Use L’Hospital’s rule to show that
\[\lim_{x \to 0}\left(\frac{1}{\sin x} - \frac{1}{x}\right) = 0\](6)
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{1}{\sin x} - \dfrac{1}{x} = \dfrac{x - \sin x}{x\sin x}\) | M1 | 3.1a |
| \(\dfrac{\frac{\mathrm{d}}{\mathrm{d}x}(x - \sin x)}{\frac{\mathrm{d}}{\mathrm{d}x}(x\sin x)} = \dfrac{1 \pm \cos x}{\sin x \pm x\cos x}\) | dM1 | 1.1b |
| \(\dfrac{\frac{\mathrm{d}}{\mathrm{d}x}(x - \sin x)}{\frac{\mathrm{d}}{\mathrm{d}x}(x\sin x)} = \dfrac{1 - \cos x}{\sin x + x\cos x}\) | A1 | 1.1b |
| \(\dfrac{\frac{\mathrm{d}}{\mathrm{d}x}(1 - \cos x)}{\frac{\mathrm{d}}{\mathrm{d}x}(\sin x + x\cos x)} = \dfrac{\pm\sin x}{\pm\cos x \pm \cos x \pm x\sin x}\) | ddM1 | 3.1a |
| \(\left(\lim\limits_{x \to 0}\left(\dfrac{1}{\sin x} - \dfrac{1}{x}\right) =\right)\lim\limits_{x \to 0}\left(\dfrac{\sin x}{2\cos x - x\sin x}\right) = \dfrac{\sin(0)}{2\cos(0) - (0)\sin(0)}\) | M1 | 1.2 |
| \(= 0\ *\) | A1* | 2.1 |
| (6) | ||
| (6 marks) |
Notes
M1: Complete method to write the function as a quotient.
dM1: Attempts differentiation of both numerator and denominator, including use of product rule with the denominator. Either numerator or denominator of the correct form. May be done separately.
A1: Fully correct differentiation of both numerator and denominator (may be separate).
ddM1: Dependent on previous method mark. Recognises the need to differentiate again and carries out the differentiation to complete the method.
M1: Clear demonstration of L’Hospital’s Rule being used to attain a limit, e.g. clear statement using limits reaching an expression that can be evaluated. Both numerator and denominator must be used appropriately. The substitution may be implied by 0/2 being reached from a suitable expression (oe if errors) but must have achieved a non-zero denominator. Not dependent, so may be scored if genuine errors in the first derivative lead to an expression that has a non-zero denominator. They may have only differentiated once.
A1*: Fully correct solution. Must see clear use of a substitution of \(x = 0\) into their derivatives. Accept as minimum e.g. \(\dfrac{0}{2 - 0}\) with each term seen evaluated or equivalent working shown. Needs to be a correct line showing substitution before reaching the printed answer with use of some limit notation. All aspects of the proof should be clear for this mark to be awarded and no errors seen.
NB Proceeding to fourth derivatives before evaluating the limit is a correct approach, and may score the final M once a limit is reached, and final A if all aspects are correct.