20N.3.AHL.TZ0.HCA_1

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to use l’Hôpital’s rule        M1

$=\underset{x\to 1}{\lim }\frac{-2x \sin \left(x^2-1\right)}{{\text{e}}^{x-1}-1}$       A1A1

Note: Award A1 for the numerator and A1 for the denominator.

substitution of $1$ into their expression        (M1)

$=\frac{0}{0}$ hence use l’Hôpital’s rule again

Note: If the first use of l’Hôpital’s rule results in an expression which is not in indeterminate form, do not award any further marks.

attempt to use product rule in numerator       M1

$=\underset{x\to 1}{\lim }\frac{-4x^2 \cos \left(x^2-1\right)-2 \sin \left(x^2-1\right)}{{\text{e}}^{x-1}}$       A1

$=-4$       A1

[7 marks]

[N/A]