IB Maths AA HL 5.13 L'Hopital's rule and limits Question Bank

Practise IB Mathematics HL 5.13 by applying l'hopital's rule and limits methods to exam-style questions.

[Maximum number: 6]

In this question you will investigate series of the form

\sum_{i=1}^{n} \boldsymbol{i}^{q}=1^{q}+2^{q}+3^{q}+\ldots+n^{q} \text { where } n, q \in \mathbb{Z}^{+}

and use various methods to find polynomials, in terms of n, for such series.
When q=1, the above series is arithmetic.

(a)

Show that $\lim _{x \rightarrow 1} f_{1}(x)$ is in indeterminate form.

[ 1 ]

$\quad \lim _{x \rightarrow 1} f_{1}(x)=\frac{n-(n+1)+1}{0}\left(=\frac{0}{0}\right)$

Note: Only award R1 for sufficient simplification of the numerator, for example, as shown above.

Do not award R1 if $\lim _{x \rightarrow 1}$ is not referred to or stated.

(b)

Hence, by applying l'Hôpital's rule, show that $\lim _{x \rightarrow 1} f_{1}(x)=\frac{1}{2} n(n+1)$ .

[ 5 ]

attempts to differentiate both the numerator and the denominator M1

\lim _{x \rightarrow 1} \frac{n(n+2) x^{n+1}-(n+1)^{2} x^{n}+1}{2(x-1)}

Note: Award A1 for $\left(\left(\lim _{x \rightarrow 1}\right) \frac{n(n+2) x^{n+1}-n(n+1) x^{n}-(n+1) x^{n}+1}{2(x-1)}\right)$ . This form can be used in subsequent work.
(l'Hôpital's rule applies again since)

\lim _{x \rightarrow 1} \frac{n(n+2) x^{n+1}-(n+1)^{2} x^{n}+1}{2(x-1)}=\frac{0}{0}

Note: Do not award R1 if $\lim _{x \rightarrow 1}$ is not referred to or stated.
Subsequent marks are independent of this R mark.
attempts to differentiate both the numerator and the denominator

\begin{aligned} & \lim _{x \rightarrow 1} \frac{n(n+2)(n+1) x^{n}-n(n+1)^{2} x^{n-1}}{2} \\ & =\frac{n(n+2)(n+1)-n(n+1)^{2}}{2}\left(=\frac{n^{3}+3 n^{2}+2 n-\left(n^{3}+2 n^{2}+n\right)}{2}\right) \\ & =\frac{n(n+1)((n+2)-(n+1))}{2}\left(=\frac{n^{2}+n}{2}\right) \\ & =\frac{1}{2} n(n+1) \end{aligned}

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AHL 5.13 (HL)—L'Hopital's rule and limits question 1

Question (a)

Question (b)