Hence or otherwise, determine the value of $\underset{x\to 0}{\lim }\left[\frac{{\left(x^2{\mathrm{e}}^x-x^2\right)}^3}{x^9}\right]$.

Show that $P_i\left(n\right)=2n\text{sin}\left(\frac{\pi }{n}\right)$.

By writing $P_c\left(n\right)$ in the form $\frac{2\text{tan}\left(\frac{\pi }{n}\right)}{\frac{1}{n}}$, find $\underset{n\to \mathrm{\infty }}{\text{lim}}{P}_c\left(n\right)$.

Find the perimeter of a regular hexagon, of side length, $x$ units, inscribed in a circle of radius 1 unit.

Use an appropriate Maclaurin series expansion to find $\underset{n\to \mathrm{\infty }}{\text{lim}}{P}_i\left(n\right)$ and interpret this result geometrically.

Hence, or otherwise, determine the value of $\underset{x\to 0}{\lim }\frac{{\text{e}}^x \cos x-1-x}{x^3}$.

Using l’Hôpital’s rule, show algebraically that the value of the limit is $-\frac{1}{4}$.

The inequality found in part (h) can be used to determine lower and upper bound approximations for the value of $\pi$.

Determine the least value for $n$ such that the lower bound and upper bound approximations are both within 0.005 of $\pi$.

Use L’Hôpital’s rule to determine the value of

$$\underset{x\to 0}{\text{lim}}\left(\frac{{{\text{e}}^{-3x}}^{{}^2}+3\text{cos}\left(2x\right)-4}{3x^2}\right)$$

Use l’Hôpital’s rule to determine the value of

$\underset{x\to 0}{\lim } \frac{2 \sin x-\sin 2x}{x^3}$.

Consider an equilateral triangle ABC of side length, $x$ units, inscribed in a circle of radius 1 unit and centre O as shown in the following diagram.

The equilateral triangle ABC can be divided into three smaller isosceles triangles, each subtending an angle of $\frac{2\pi }{3}$ at O, as shown in the following diagram.

Using right-angled trigonometry or otherwise, show that the perimeter of the equilateral triangle ABC is equal to $3\sqrt{3}$ units.

Interpret your answer to part (e)(i) geometrically.

Use l’Hôpital’s rule to find

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

Show that $P_c\left(n\right)=2n\text{tan}\left(\frac{\pi }{n}\right)$.

Hence find $\underset{x\to 0}{\text{lim}}\left(\frac{{\int }_0^x\left({{\text{e}}^{-3t}}^{{}^2}+3\text{cos}\left(2t\right)-4\right)\text{d}t}{{\int }_0^{x}3t^2\text{d}t}\right)$.

Use l’Hôpital’s rule to determine the value of $\underset{x\to 0}{\lim }\left(\frac{2x \cos \left(x^2\right)}{5 \tan x}\right)$.

Hence or otherwise, find $\underset{x\to 0}{\text{lim}}\frac{\text{arcsin}\left(2x\right)-2x}{{\left(2x\right)}^3}$.

Use l’Hôpital’s rule to find $\underset{x\to 0}{\lim }\left(\frac{\text{arctan} 2x}{\tan 3x}\right)$.

Show that a finite limit only exists for $k=\frac{\mathrm{\pi }}{4}$.

Prove by mathematical induction that $\frac{d^n}{d{x}^n}\left(x^2{\mathrm{e}}^x\right)=\left[x^2+2nx+n\left(n-1\right)\right]{\mathrm{e}}^x$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Using L’Hôpital’s rule, find $\underset{x\to 0}{\text{lim}}\left(\frac{\text{tan}3x-3\text{tan}x}{\text{sin}3x-3\text{sin}x}\right)$.

Use the results from part (d) and part (f) to determine an inequality for the value of $\pi$ in terms of $n$.

By considering limits, show that the graph of $y=f\left(x\right)$ has a horizontal asymptote and state its equation.

Use l’Hôpital’s rule to determine the value of

$$\underset{x\to 0}{lim}\frac{{\sin }^{2}x}{x\ln (1+x)}.$$

Use the Maclaurin series for $\tan x$ to find $\underset{n\to \infty }{\lim }\left(4n \tan \frac{\mathrm{\pi }}{n}\right)$.

By using the Maclaurin series for $\text{arctan} x$ and the result from part (b), find $\underset{x\to 0}{\lim }\left(\frac{x\text{\hspace{0.05em}arctan} 2x}{\text{sec} x-1}\right)$.

Hence or otherwise, determine the Maclaurin series of $f\left(x\right)=x^2{\mathrm{e}}^x$ in ascending powers of $x$, up to and including the term in $x^4$.