Topic 4 - Limits and Series

Week 11 Material

Question 4.1 $\;$ Find the limits of the following sequences as $n\rightarrow\infty$ or say why no limit exists.

$\;\;(a)\;\; s_n=\dfrac{2n+1}{n+2},\hspace{1cm}$ $\;\;(b)\;\; s_n=\dfrac{n+(-1)^n}{n},\hspace{1cm}$ $\;\;(c)\;\; s_n=\dfrac{n^5}{17n^4+12},$

$\;\;(d)\;\; s_n=\cos(n\pi),\hspace{1cm}$ $\;\;(e)\;\; s_n=\dfrac{\sin(n\pi/2)}{n},\hspace{1cm}$ $\;\;(f)^*\;\; s_n=\dfrac{7+2^n}{8+2^{n+1}}.$

Question 4.2 $\;$

$\;\;(a)\;\;$ Let $c>1$ be a constant. Show that the sequence $s_n=c^n$ is unbounded as $n\rightarrow\infty$.

$\qquad\;\,$ Hint: look at what happens when $n\geq\dfrac{\ln M}{\ln c}$ for large $M$.

$\;\;(b)\;\;$ Find the limit of the sequence $s_n=\left(\dfrac{2n+1}{n+1}\right)^{2n}$ as $n\rightarrow\infty$ or carefully explain why it

$\qquad\;\,$ doesn’t exist. Hint: first show that $\dfrac{2n+1}{n+1}\geq\dfrac32$ for $n\geq 1$.

Question 4.3 $\!{}^*\;$ Show that $0\leq \dfrac{2^n}{n!} \leq \dfrac{4}{n}$ for all $n\geq 1$. Use this to find the limit of $\dfrac{2^n}{n!}$ as $n\rightarrow\infty$.

Question 4.4 $\;$ $\;\;(a)\;\;$ By considering an area under the graph $y=1/x$, show that for $n\geq 1$, we have \[\frac{1}{n+1}\leq\ln\left(1+\frac{1}{n}\right)\leq\frac{1}{n}.\]

$\;\;(b)\;\;$ Conclude that $\left(1+\frac1n\right)^n\leq e\leq\left(1+\frac1n\right)^{n+1}$ and hence express $e$ as the limit of a sequence.

Question 4.5 $\;$ Evaluate the following limits of functions or say why no limit exists.

$\;\;(a)\;\;\displaystyle\lim_{x\rightarrow -5}\;\dfrac{x^2}{5-x},\hspace{1.8cm}$ $\;\;(b)\;\;\displaystyle\lim_{x\rightarrow 2}\;\dfrac{1/x-1/2}{x-2}\hspace{1.7cm}$ $\;\;(c)\;\;\displaystyle\lim_{x\rightarrow 0}\;(2x-8)^{1/3},$

$\;\;(d)^*\;\;\displaystyle\lim_{x\rightarrow 9}\;\dfrac{x-9}{\sqrt{x}-3}\hspace{1.8cm}$ $\;\;(e)\;\;\displaystyle\lim_{x\rightarrow 1}\;\dfrac{x^3-1}{x^2-1}\hspace{2.4cm}$ $\;\;(f)\;\;\displaystyle\lim_{x\rightarrow 3}\;\dfrac{x^2+x+12}{x-3},$

$\;\;(g)\;\;\displaystyle\lim_{x\rightarrow 3}\;\dfrac{x^2+x-12}{x-3}\hspace{0.9cm}$ $\;\;(h)\;\;\displaystyle\lim_{x\rightarrow 3}\;\dfrac{(x^2+x-12)^2}{x-3}\hspace{0.8cm}$ $\;\;(i)\;\;\displaystyle\lim_{x\rightarrow 3}\;\dfrac{x^2+x-12}{(x-3)^2}.$

Question 4.6 $\;$ Using the standard limit $\displaystyle\lim_{x\rightarrow 0}\dfrac{\sin{x}}{x}=1$, evaluate

$\;\;(a)\;\;\displaystyle\lim_{x\rightarrow 0}\;\dfrac{\sin(4x)}{3x},\hspace{1.1cm}$ $\;\;(b)^*\;\;\displaystyle\lim_{x\rightarrow 0}\;\dfrac{\sin^2(4x)}{3x^2},\hspace{1.1cm}$ $\;\;(c)\;\;\displaystyle\lim_{x\rightarrow 0}\;\dfrac{\cos(2x)-1}{x},$

$\;\;(d)\;\;\displaystyle\lim_{x\rightarrow 0}\; x\cot(3x),\hspace{1cm}$ $\;\;(e)\;\;\displaystyle\lim_{x\rightarrow 0}\;\dfrac{1-\cos{x}}{x^2},\hspace{1cm}$ $\;\;(f)\;\;\displaystyle\lim_{x\rightarrow \pi/2}\;\dfrac{\cos{x}}{2x-\pi}.$

Question 4.7 $\;$ Evaluate the limits as $x\rightarrow\infty$ of the following functions:

$\;\;(a)\;\;\dfrac{2x+1000}{x+2},\hspace{1cm}$ $\;\;(b)\;\;\dfrac{3x^3+7}{4x^3+2x^2+3x+1},\hspace{1cm}$ $\;\;(c)^*\;\;\sqrt{x^2+x}-x.$

Week 12 Material

Question 4.8 $\;$ Find the points where the following functions are not continuous:

$\;\;(a)\;\;\dfrac{x^4+5x+7}{x^2-7x+10},\hspace{1cm}$ $\;\;(b)\;\;\sec{x}=\dfrac{1}{\cos{x}},\hspace{1cm}$ $\;\;(c)\;\;\dfrac{\tan{x}}{x^2-x}.$

Question 4.9 $\!{}^*\;$ Sketch the following function and determine where it is not continuous: \[f(x)=\begin{cases} 2x+1 &\quad\text{for $x\leq 0$,} \\ 1 &\quad\text{for $0<x\leq 1$,} \\ x^2+1 &\quad\text{for $x>1$.} \end{cases}\]

Question 4.10 $\;$ The function $f(x)=\dfrac{\sqrt{x+3}-2}{x-1}$ isn’t continuous at $x=1$ since it isn’t defined there. Modify it by defining $f(1)$ so that the function becomes continuous for all $x\geq 0$.

Question 4.11 $\;$ By using the definition $f'(x)=\displaystyle\lim_{h\rightarrow 0}\; \frac{f(x+h)-f(x)}{h}$, find the derivative of:

$\;\;(a)\;\;f(x)=x^3,\hspace{2.75cm}$ $\;\;(b)\;\;f(x)=x^n\;\;$ for $n\in\mathbb{N},\;\;$ (Hint: use the Binomial Theorem)

$\;\;(c)^*\;\;f(x)=x^{-1}\;\;$ for $x\neq 0,\hspace{0.3cm}$ $\;\;(d)\;\;f(x)=\sqrt{x}\;\;$ for $x>0$,

$\;\;(e)\;\;f(x)=\cos{x},\hspace{2.25cm}$ $\;\;(f)\;\;f(x)=1/\sqrt{1+x}\;\;$ for $x>-1.$

Question 4.12 $\;$ What can you say about the limit \[\displaystyle\lim_{h\rightarrow 0}\; \frac{f(x+h)-f(x)}{h}\] for $f(x)=x^{1/3}$ in the special case $x=0$? What does this tell us about the graph of $f(x)$?

Question 4.13 $\!{}^*\;$ A function $f(x)$ is defined by $f(x)=\begin{cases} (x-1)^2 &\;\text{for $x\leq 1$,} \\ (x-1)^4 &\;\text{for $x\geq 1$}. \end{cases}$

How many times is it differentiable at $x=1$?

Question 4.14 $\;$ Suppose $f(x)$ is continuous at $x=0$. Show that the function $g(x)=f(x)\sin{x}$ is differentiable at $x=0$ and find the derivative $g'(0)$.

Question 4.15 $\;$ Use the Intermediate Value Theorem to show that $p(x)=x^4-2x-1$ has at least two roots in the interval $-1<x<2$.

Question 4.16 $\;$ Suppose we have a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ where $n$ is odd. By considering the limits of $p(x)$ as $x\rightarrow\pm\infty$, show that $p(x)=0$ has a real solution. Why is this different to the case where $n$ is even?

Question 4.17 $\;$ Suppose a quadratic polynomial $p(x)=ax^2+bx+c$ has a local minimum at $x=2$ and that $p(-1)=3$ and $p(3)=-1$. Find $a$, $b$ and $c$. Repeat the same question but with a local maximum at $x=2$.

Week 13 Material

Question 4.18 $\;$ Use l’Hôpital’s Rule (in its “$0/0$” form) to evaluate the following limits:

$\;(a)\;\displaystyle\lim_{x\rightarrow 0}\,\dfrac{\ln (1+x)-x}{1-\cos{x}},\hspace{0.7cm}$ $\;(b)\;\displaystyle\lim_{x\rightarrow 0}\,\dfrac{\cosh{x}-\cos{x}}{x^2},\hspace{0.9cm}$ $\;(c)^*\;\displaystyle\lim_{x\rightarrow 0}\,\dfrac{\tan{x}-x}{x-\sin{x}},$

$\;(d)\;\displaystyle\lim_{x\rightarrow 0}\,\dfrac{\arcsin{x}}{x},\hspace{1.7cm}$ $\;(e)\;\displaystyle\lim_{x\rightarrow 0}\,\dfrac{\sqrt{5+2x}-\sqrt{5}}{x},\hspace{0.8cm}$ $\;(f)\;\displaystyle\lim_{x\rightarrow 0}\,\dfrac{\arcsin{x}}{\sin{x}},$

$\;(g)\;\displaystyle\lim_{x\rightarrow \pi/2}\,\dfrac{\cos{x}}{\pi-2x},\hspace{1.5cm}$ $\;(h)\;\displaystyle\lim_{x\rightarrow 1}\,\dfrac{\ln{x}}{x^2-1},\hspace{2.3cm}$ $\;(i)\;\displaystyle\lim_{x\rightarrow 0}\,\dfrac{\sin{x}+x}{x+x^2},$

$\;(j)\;\displaystyle\lim_{x\rightarrow \pi/2}\,\dfrac{\tan(5x)}{\tan{x}},\hspace{1.4cm}$ $\;(k)\;\displaystyle\lim_{x\rightarrow\infty}\,\sqrt{x^2+x+1}-\sqrt{x^2+1}.$

Question 4.19 $\;$ Use l’Hôpital’s Rule (in its “$\infty/\infty$” form) to evaluate the following limits:

$\;(a)\;\displaystyle\lim_{x\rightarrow\infty}\, \dfrac{3x^3+7}{4x^3+2x^2+3x+1},\hspace{0.95cm}$ $\;(b)\;\displaystyle\lim_{x\rightarrow\infty}\, \dfrac{\ln{x}}{x^2},\hspace{2.8cm}$ $\;(c)^*\;\displaystyle\lim_{x\rightarrow\infty}\, \dfrac{x^2}{e^x},$

$\;(d)\;\displaystyle\lim_{x\rightarrow\infty}\, \dfrac{\ln{x}}{x^p} \;\;$ for $p>0,\hspace{1.7cm}$ $\;(e)\;\displaystyle\lim_{x\rightarrow \infty}\, \dfrac{x^p}{e^x}\;\;$ for $p>0.\hspace{0.9cm}$

Question 4.20 $\;$ Use Leibniz’s Rule to find the third derivative of the following functions

$\;(a)^*\;f(x)=x^4\ln{x},\hspace{1.5cm}$ $\;(b)\;f(x)=e^{-2x}\cos (3x),\hspace{1.5cm}$ $\;(c)\;f(x)=(x-1)^4\sin (2x).$

Question 4.21 $\;$ For the function $y(x)=e^{-x^2}$, find the derivative $y'(x)$ as a simple product involving $x$ and $y$. Hence apply Leibniz’s Rule to show that for $n\geq 1$, \[y^{(n+1)}+2xy^{(n)}+2ny^{(n-1)}=0\] where $y^{(n)}$ denotes the $n$th derivative of $y$.

Question 4.22 $\;$ Use the Newton-Raphson method to find, correct to three significant figures, the positive real solution of each of the following:

$\;\;(a)^*\;\;$ $x^4+x-3=0$ starting with $x_0=1$. What happens if you start at $x_0=-1$ instead?

$\;\;(b)\;\;$ $e^{2x}-x=2$ starting with $x_0=0.5$,

$\;\;(c)\;\;$ $1000x^3-x-1=0$ starting with $x_0=0.1$. What happens if you start at $x_0=0$ instead?

Question 4.23 $\;$ The Lambert function $W(a)$ for $a\geq 0$ gives the value $W(a)=w$ such that $we^w=a$ and has applications in many areas such as quantum mechanics and biochemistry. It can not be expressed nicely in terms of other more familiar functions, but can be approximated by a sequence $w_0, w_1,...$ tending to $w$. Use the Newton-Raphson method to find an iteration that improves an approximation $w_{k}$ to $w_{k+1}$.

With a calculator, find $W(2)$ to 3 decimal places, starting from an initial guess $w_0=1$.

Week 14 Material

Question 4.24 $\;$ By looking at the Taylor series $\sin{x}=x-x^3/6+\cdots$, it’s possible to show that \[x-\frac{x^3}{6}<\sin{x}<x\qquad\text{for $x>0$.}\] Use this to find approximately the range of values of $x>0$ for which

$\;(a)\; 1-\dfrac{\sin{x}}{x}<10^{-2},\hspace{1cm}$ $\;(b)\; 1-\dfrac{\sin{x}}{x}<10^{-4},\hspace{1cm}$ $\;(c)\; 1-\dfrac{\sin{x}}{x}<10^{-10}.$

Hence guess the limit of $\;\dfrac{\sin{x}}{x}\;$ as $x\rightarrow 0$. (You should already know the answer!)

Question 4.25 $\;$ Use the definition of Taylor polynomials to find:

$\;(a)\;$ the degree 3 Taylor polynomial of $f(x)=\cosh{x}$ about $x=0$.

$\;(b)\;$ the degree 3 Taylor polynomial of $f(x)=\tan{x}$ about $x=0$.

$\;(c)\;$ the degree 3 Taylor polynomial of $f(x)=e^{x/2}$ about $x=3$.

$\;(d)^*\;$ the degree 3 Taylor polynomial of $f(x)=\sqrt{x}$ about $x=a>0$. What happens if you try to expand about $x=0$?

Question 4.26 $\;$ $\;(a)\;$ Show that the Taylor expansion for $f(x)=\dfrac{1}{1-x}$ (valid for $|x|<1$) is \[\frac{1}{1-x}=1+x+x^2+x^3+\cdots\] $\;(b)\;$ Use this to find a closed formula for the series $\;x+2x^2+3x^3+4x^4+\cdots$

Question 4.27 $\;$ Use the Taylor series about $x=0$ for $\cos{x}$, $\sin{x}$ and $\ln(1+x)$ to find the degree 4 Taylor polynomial about $x=0$ of the following functions:

$\;(a)\;\;\cos(2x),\hspace{3cm}$ $\;(b)\;\;\sin^2x,\hspace{3cm}$ $\;(c)\;\;\ln(2+x^2).$

Question 4.28 $\;$ The Sine Integral $\operatorname{Si}(x)$ is defined by \[\operatorname{Si}(x)=\int_0^{x}\;\frac{\sin{t}}{t}\;dt\] and has applications in areas such as Signal Processing. Find the Taylor series for $\operatorname{Si}(x)$ about $x=0$.

Question 4.29 $\;$ Use standard Taylor series expansions from the table in the lecture notes to evaluate the limits of the following functions as $x\rightarrow 0$.

$\;(a)\;\;\dfrac{\ln (1+x)-x}{1-\cos{x}},\hspace{2cm}$ $\;(b)\;\;\dfrac{\cosh{x}-\cos{x}}{x^2},\hspace{2cm}$ $\;(c)^*\;\;\dfrac{\tan{x}-x}{x-\sin{x}},$

$\;(d)\;\;\dfrac{\arcsin{x}}{x},\hspace{3.05cm}$ $\;(e)\;\;\dfrac{\sqrt{5+2x}-\sqrt{5}}{x},\hspace{1.8cm}$ $\;(f)\;\;\dfrac{\arcsin{x}}{\sin{x}},$

$\;(g)\;\;\dfrac{1}{x^3} \left(\dfrac{1}{\sin{x}}-\dfrac{1}{x}-\dfrac{x}{6}\right),\hspace{0.7cm}$ $\;(h)\;\;\dfrac{\exp\left(\dfrac{\sin{x}}{1-3x}\right)-x-1}{x\cos{x}-\ln(1+x)}.\;$

(These last two might take serious effort but would be virtually impossible using L’Hôpital’s Rule!)

Question 4.30 $\;$ $\;(a)\;$ Let $f(x)=x/\sqrt{1+x^2}$. Show that \[(1+x^2)^2f''(x)+3f(x)=0\] and use this to find the degree 3 Taylor polynomial for $f(x)$ about $x=0$.

$\;(b)\;$ Let $f(x)=(\sinh^{-1}x)/\sqrt{1+x^2}$. Show that \[(1+x^2)f'(x)+f(x)=1\] and use this to find the degree 3 Taylor polynomial for $f(x)$ about $x=0$.

Question 4.31 $\;$ $\;(a)\;$ Use a Taylor polynomial for $\cos{x}$ and the bound on the remainder term to estimate $\cos(1/2)$ to within 0.01, without using a calculator.

$\;(b)^*\;$ Use a Taylor polynomial for $e^x$ and the bound on the remainder term to estimate $e^{1/2}$ to within 0.01 without using a calculator. (You may use the fact that $e<4$, which surprisingly is useful here.)

Question 4.32 $\;$ Estimate the maximum error between $f(x)=\sin{x}$ and its degree 3 Taylor polynomial about $x=0$ for $x$ in the interval $-\pi/4\leq x\leq\pi/4$. How large is the error when $\cos{x}$ is estimated in the same way?