Evaluate limit using series

4

1. Write down the expansion of \(\sin 3 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
2. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { 3 x \cos 2 x - \sin 3 x } { 5 x ^ { 3 } } \right]$$

2

1. Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\).
2. Show that, for some value of \(k\), $$\lim _ { x \rightarrow 0 } \left[ \frac { 2 x - \sin 2 x } { x ^ { 2 } \ln ( 1 + k x ) } \right] = 16$$ and state this value of \(k\).

3

1. 1. Write down the expansion of \(\ln ( 1 + 2 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\).
   2. Hence, or otherwise, find the first two non-zero terms in the expansion of $$\ln [ ( 1 + 2 x ) ( 1 - 2 x ) ]$$ in ascending powers of \(x\) and state the range of values of \(x\) for which the expansion is valid.
2. Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 3 x - x \sqrt { 9 + x } } { \ln [ ( 1 + 2 x ) ( 1 - 2 x ) ] } \right]\).

1. Use l'Hospital's Rule to show that

$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \left( e ^ { \sin x } - \cos ( 3 x ) - e \right) } { \tan ( 2 x ) } = - \frac { 3 } { 2 }$$

1. Use L'Hospital's rule to show that

$$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { \sin x } - \frac { 1 } { x } \right) = 0$$ (6)

9 Use l'Hôpital's rule to show that $$\lim _ { x \rightarrow \infty } \left( x \mathrm { e } ^ { - x } \right) = 0$$ Fully justify your answer.
[0pt] [4 marks]
\(11 \quad\) The line \(L _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 2
2
3 \end{array} \right] + \lambda \left[ \begin{array} { c } 2
3
- 1 \end{array} \right]\)
The line \(L _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 6
4
1 \end{array} \right] + \mu \left[ \begin{array} { c } - 2
1
1 \end{array} \right]\)

13 Use l'Hôpital's rule to prove that $$\lim _ { x \rightarrow \pi } \left( \frac { x \sin 2 x } { \cos \left( \frac { x } { 2 } \right) } \right) = - 4 \pi$$

4 Which one of the following statements is correct?
Tick ( ✓ ) one box.
\(\lim _ { x \rightarrow 0 } \left( x ^ { 2 } \ln x \right) = 0\) □
\(\lim _ { x \rightarrow 0 } \left( x ^ { 2 } \ln x \right) = 1\)
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-03_110_108_1238_991}
\(\lim _ { x \rightarrow 0 } \left( x ^ { 2 } \ln x \right) = 2\) □
\(\lim _ { x \rightarrow 0 } \left( x ^ { 2 } \ln x \right)\) is not defined.
\includegraphics[max width=\textwidth, alt={}, center]{9a2f64fb-71d1-4140-b701-c9fbb5b3891c-03_106_108_1564_991}

11

1. Starting from the series given in the formulae booklet, show that the general term of the Maclaurin series for $$\frac { \sin x } { x } - \cos x$$ is $$( - 1 ) ^ { r + 1 } \frac { 2 r } { ( 2 r + 1 ) ! } x ^ { 2 r }$$ 11
2. Show that $$\lim _ { x \rightarrow 0 } \left[ \frac { \frac { \sin x } { x } - \cos x } { 1 - \cos x } \right] = \frac { 2 } { 3 }$$

2 Which one of the expressions below is not equal to zero?
Circle your answer.
[0pt] [1 mark]
\(\lim _ { x \rightarrow \infty } \left( x ^ { 2 } \mathrm { e } ^ { - x } \right)\)
\(\lim _ { x \rightarrow 0 } \left( x ^ { 5 } \ln x \right)\)
\(\lim _ { x \rightarrow \infty } \left( \frac { \mathrm { e } ^ { x } } { x ^ { 5 } } \right)\)
\(\lim _ { x \rightarrow 0 } \left( x ^ { 3 } \mathrm { e } ^ { x } \right)\)