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 $-\frac{1}{4}$.

Consider a square of side length, $x$ units, inscribed in a circle of radius 1 unit. By dividing the inscribed square into four isosceles triangles, find the exact perimeter of the inscribed square.

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.

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 $x^4$.