Limit using series expansion

A question is this type if and only if it asks to find a limit (typically as x→0) by using Taylor/Maclaurin series expansions to simplify indeterminate forms.

4 questions · Challenging +1.0

4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n

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AQA FP3 2010 January Q4

5 marks Standard +0.8

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

AQA FP3 2012 June Q2

5 marks Standard +0.8

Write down the expansion of $\sin 2 x$ in ascending powers of $x$ up to and including the term in $x ^ { 5 }$.
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 $k$.

AQA FP3 2015 June Q3

8 marks Challenging +1.2

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.
Find $\lim _ { x \rightarrow 0 } \left[ \frac { 3 x - x \sqrt { 9 + x } } { \ln [ ( 1 + 2 x ) ( 1 - 2 x ) ] } \right]$.

AQA Further Paper 2 2020 June Q11

8 marks Challenging +1.2

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 $$(-1)^{r+1} \frac{2r}{(2r + 1)!} x^{2r}$$ [4 marks]
Show that $$\lim_{x \to 0} \left[ \frac{\sin x}{x} - \cos x \right] \frac{1}{1 - \cos x} = \frac{2}{3}$$ [4 marks]