W1 Limits and Continuity

NOTATIONS

Standard Abbreviations

Symbol	What it means
|	such that/given that
$\forall$	for all
$\exists$	there exists
$\equiv$	equivalent to
$\textmd{i.e}$	that is
$\approx$	approximate
$\ll$	much smaller than

Standard Notation for Set of Numbers

Symbol	What it means
$\mathbb{N}$	Natural/Counting Numbers
$\mathbb{Z}$	Integers
$\mathbb{Q}$	Rational Numbers (can be expressed in fraction)
$\mathbb{R}$	Real Numbers
$\mathbb{C}$	Complex Numbers
$\mathbb{R^2}$	2 dimentional plane (x,y)
$\mathbb{R^3}$	3 dimensional plane (x,y,z)

Standard Notation for Intervals

Symbol	What it means
$\in$	element of
$(a,b)$	open interval
$[a,b]$	closed interval
$[a,b)$ $(a,b]$	partial open closed interval
\	not including

More Standard Notations

• Logarithms
  • natural logarithm: $\log x, \log_ex, \ln x$
  • base 10 logarithm: $\log_{10} x, \textmd{ld } x$
  • base 2 logarithm: $\log_2x, \textmd{lb } x$
• Inverse Trigonometric Functions
  • $\arcsin x, \sin^{-1}x$
  • $\arccos x, \cos^{-1}x$
  • $\arctan x, \tan^{-1}x$
  • $\textmd{arccsc } x, \csc^{-1}x$
  • $\textmd{arcsec } x, \sec^{-1}x$
  • $\textmd{arccot } x, \cot^{-1}x$
• $\implies$ implies
• $\iff$ if and only if (iff)
• $\rightarrow$ approaches

LIMITS

Definition of limit of a function

• $\lim_{x\rightarrow a} f(x)=L$
• in words: limit of $f(x)$ as $x$ approaches $a$ is $L$
• $\lim_{x\rightarrow a} f(x)$ is not necessarily = to $f(a)$

$\infty$ or $-\infty$ is NOT a real number

• N E V E R write $\lim_{x\rightarrow a} f(x)=\pm\infty$ (you will lose marks)
• example $f(x)=\frac{1}{x^2}$, evaluate $\lim_{x\rightarrow 0} f(x)$
  • $f(x)$ is unbounded as $x$ approaches $0$. Hence, $f(x)$ cannot be arbitrarily close to any real number as $x$ approaches 0
  • This shows that $\lim_{x\rightarrow 0}f(x)$ DOES NOT EXIST

THEOREM limit of $f(x)$ only exists if the right-hand limit and left-hand limit is the same

$\boxed{\exists\lim_{x\rightarrow a}f(x)=L\iff \lim_{x\rightarrow a^-}f(x)=\lim_{x\rightarrow a^+}f(x)}$

LIMIT LAWS

• Let $\lim_{x\rightarrow a}f(x)=M$, $\lim_{x\rightarrow a}g(x)=N$

#	Rule Name	Rule
1	Sum/Difference Rule	$\lim_{x\rightarrow a}[f(x)\pm g(x)] = M\pm N$
2	Constant Multiple Rule	$\lim_{x\rightarrow a}[cf(x)]=cM$
3	Product Rule	$\lim_{x\rightarrow a}[f(x)g(x)]=M\cdot N$
4	Quotient Rule	$\lim_{x\rightarrow a}[\frac{f(x)}{g(x)}]=\frac{M}{N}$
5	Constant Rule	$\lim_{x\rightarrow a}c=c$
6	Power Rule I	$\lim_{x\rightarrow a}x^n=a^n$ $\lim_{x\rightarrow a}x=a$
7	Power Rule II	$\lim_{x\rightarrow a}[f(x)]^n=M^n$
8	Root Rule	$\lim_{x\rightarrow a}[\sqrt[n]{f(x)}]=\sqrt[n]{M}$

LIMIT TRICKS

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STANDARD LIMITS TO INFINITY

1. $\lim_{x\rightarrow\infty}\frac{1}{x^p}=0, p>0$
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2. $\lim_{x\rightarrow\infty}r^x=0, 0\le r <1$
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3. $\lim_{x\rightarrow\infty}\frac{1}{e^{ax}}=0, a>0$
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Terminology: Diverge/Converge

• If $\lim_{x\rightarrow a}f(x)$ exists, we say that $f(x)$ converges as $x$ approaches $a$
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• If $\lim_{x\rightarrow a}f(x)$ does not exist, we say that $f(x)$ diverges as $x$ approaches $a$
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SANDWICH THEOREM

• The Organic Chemistry Tutor
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FUNCTION GRAPHS (Review)

• to help easily figure out standard limits

Line Tests

• vertical line test: check if graph is function
• horisontal line test: check if graph is one-to-one or many-to-one

Straight Lines

• $\boxed{y=mx+c}$
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  • Gradient: $m=\frac{y_2-y_1}{x_2-x_1}$
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  • $y$-intercept: $c$
  • Angle of inclination: $m=\tan\theta, 0\degree\le\theta<180\degree$
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  • Parallel: $m_1=m_2$
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  • Perpendicular: $m_1m_2=-1$
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Parabolas/Quadratic

• $\boxed{SF: y=ax^2+bx+c}$
  • $y$-intercept: $c$
  • $a>0$:
  • $a<0$:
• $\boxed{VF: y=a(x-h)^2+k}$
  • Vertex: $(-h,k)$
  • Axis of symmetry: $x=-h$
• $\boxed{FF: y=a(x-p)(x-q)}$
  • $x$-intercepts: $p,q$
• Quadratic Formula
  • $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Cubics

• $\boxed{y=ax^3+bx^2+cx+d}$
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Absolute Value

• $\boxed{y=|x|}$
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Hyperbolas

• $\boxed{y=\frac{1}{x}}$
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Exponential Functions

• $\boxed{y=a^x, \textmd{where } a>0}$
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Logarithmic Functions

• $\boxed{y=\log_ax,\textmd{where }a>1}$
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Semicircles

• $\boxed{y=\pm\sqrt{r^2-x^2}}$
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Square Root Graph

• $\boxed{y=\sqrt{x},\textmd{where }x\ge0}$
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CONTINUITY

Definition of continuity

• function $f$ is continuous at $x=a$ if: $\boxed{\lim_{x\rightarrow a}f(x)=f(a)}$

Continuity Theorem 1

• if $f$ and $g$ are continuous at $x=a$ then these are also continuous at $x=a$:

1. $f+g$
2. $cf$
3. $fg$
4. $\frac{f}{g}\bold{\textmd{ if }g(x)\ne0}$

Contiuity Theorem 2

• if $f$ and $g$ are continuous at $x=a$ then $g \circ f$ (aka $g(f(x))$) is continuous

Continuity Theorem 3

• The following function types are continuous at every point in their domains:

1. polynomials
2. trigonometric functions
   • $\sin(x), \cos(x)$
   • $\tan(x), \sec(x); x\ne\pm\frac{\pi}{2},\pm\frac{3\pi}{2}, ...$
   • $\cosec(x),\cot(x); x\ne\pm\pi,\pm2\pi$
   • $\arcsin(x),\arccos(x);x\leftarrow[-1,1]$
   • $\arctan$
3. exponential functions
   • $a^x$ for $a>0$
4. logarithm functions
   • $\log_ax$ for $a>0,x>0$
5. $n^{th}$ root functions
   • $\sqrt[n]{x}$ for $n\in\{2,3,4,...\},x\ge0$
6. hyperbolic functions
   • $\sinh(x),\cosh(x),\tanh(x),\textmd{sech}(x)$
   • $\textmd{cosech}(x),\textmd{coth}(x)\textmd{arccosh}(x); x\ne0$
   • $\textmd{arcsinh}(x)$
   • $\textmd{arctanh}(x), x\in(-1,1)$
   • $\textmd{arccosh}(x); x\ge1$
   • $\textmd{arcsech}(x); x\le1$
   • $\textmd{arccoth}(x); x\in(-\infty,-1)\cup(1,\infty)$
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