\( % theorem \newenvironment{pf}{\textbf{Proof.}}{\rule{1ex}{1ex}} \newtheorem{qu}{Question} \newtheorem{thm}{Theorem} \newtheorem{rmk}{Remark} \newtheorem{go}{Goal} \newtheorem{df}{Definition} \newtheorem{prop}{Proposition} \newtheorem{mot}{Motivation} \newtheorem{ex}{Example} \lstnewenvironment{code}{}{} \newcommand{\nc}{\newcommand} %% env \nc{\env}[2]{ \begin{#1} #2 \end{#1} } \nc{\arr}[2]{ \begin{array}{#1} #2 \end{array} } \nc{\arrb}[2]{ \left\{ \begin{array}{#1} #2 \end{array} \right. } \nc{FTC}[3]{ \left[#3\right]^{#2}_{#1} } \nc{\bracs}[1]{ \left[#1\right] } \nc{\brac}[1]{ \left(#1\right) } \nc{\bracp}[1]{ \left\{#1\right\} } \nc{\intd}[4]{\int^{#2}_{#1}\left(#3\right) #4} % shortcut \nc{\Lra}{\Leftrightarrow} \nc{\Ra}{\Rightarrow} \nc{\La}{\Leftarrow} \nc{\C}{\mathbb{C}} \nc{\R}{\mathbb{R}} \nc{\Fc}{\mathcal{C}} \nc{\bds}{\boldsymbol} \nc{\ds}{\displaystyle} % begin end \nc{\bit}{\begin{itemize}} \nc{\eit}{\end{itemize}} \nc{\bcode}{\begin{code}} \nc{\ecode}{\end{code}} \nc{\bsage}{\begin{sageblock}} \nc{\esage}{\end{sageblock}} %pic \nc{\pic}{\includegraphics} \nc{\svg}{\includesvg} \nc{\pica}{\includegraphics[width=100px,height=100px]} \nc{\picb}{\includegraphics[width=150px,height=150px]} \nc{\vs}{\vspace} \nc{\tbf}{\textbf} %arrow \nc{\upa}{\nearrow} \nc{\downa}{\searrow} \nc{\upw}{\rcurvearrowright} \nc{\downw}{\curvearrowright} \)

MA441- LEC 3 - The limit of a function II

Goal

  • one-sided limits
  • inifite limits
  • vertical asymtote(s)

Example 1. (Jump)

Let \( f(x)=\frac{|x|}{x} \). Find \(\lim_{x\to 0} f(x)\)

Therefore there is no limit at \(x=0\).

But we can define one-sided limits.

Left-handed limit

\[\lim_{x\to x_0^-} f(x)=L\]

we say the limit of \(f(x)\) as \(x\) approaches \(x_0\) from the left is equal to \(L\) if we can make the values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\)'s to be sufficiently close to \(x_ 0\) and \(x\) is less than \(x_0\).

right-handed limit

\[\lim_{x\to x_0^+} f(x)=L\]

we say the limit of \(f(x)\) as \(x\) approaches \(x_0\) from the right is equal to \(L\) if we can make the values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\)'s to be sufficiently close to \(x_0 \) and \(x\) is greater than \(x_0\).

Now let's find
\( \lim_{x\to 0^-} f(x)=? \) and \( \lim_{x\to 0^+} f(x)=? \)

proposition.

\[ \lim_{x\to x_0} f(x)=L \]

is equivalent to

\( \displaystyle \lim_{x\to x_0^-} f(x)=L \) and \( \displaystyle \lim_{x\to x_0^+} f(x)=L \)

Why do we need a rigorous definition of the limit of a funciton.

Let \(f(x)=x^2\cdot cos(\frac{\pi}{x})\). Find \(\lim_{x\to 0} f(x)\).

Check up

Check

  • \(f(-4)=?~\), \(\displaystyle \lim_{x\to -4^-} f(x)=?~ \), \(\displaystyle \lim_{x\to {-4}^+} f(x)=?~\), \(\displaystyle \lim_{x\to {-4}} f(x)=?~\)

  • \( f(1)=?~\), \(\displaystyle \lim_ {x\to 1^-} f(x)=?~ \), \(\displaystyle \lim_ {x\to 1^+} f(x)=?~ \), \(\displaystyle \lim_ {x\to 1} f(x)=?~ \)

  • \(f(6)=?~\), \(\displaystyle \lim_ {x\to 6^-} f(x)=?~ \), \(\displaystyle \lim_ {x\to 6^+} f(x)=?~ \), \(\displaystyle \lim_ {x\to 6} f(x)=?~ \)

infinite limits

Example.

Find \( lim_{x\to 0} \frac{1}{x^2} \) if it exists.

1/x^2
From the graph, \[ \lim_{x\to 0} \frac{1}{x^2}=\infty\]

Example.

Find \( lim_{x\to 0} \frac{1}{x} \) if it exists.

1/x

From the graph, \[ \lim_{x\to 0^+} \frac{1}{x}= \infty ~~ lim_{x\to 0^-} \frac{1}{x}=- \infty \] Therefore, the limit does not exist at \(x=0\).

Vertical asymptote \( x=a \) of the curve \( y=f(x) \).

if at least one of the following statements is true:

  • \[ \lim_ {x\to a^+} f(x)=\infty \]
  • \[ \lim_ {x\to a^+} f(x)=-\infty \]
  • \[ \lim_ {x\to a^-} f(x)=\infty \]
  • \[ \lim_ {x\to a^-} f(x)=-\infty \]

Example.

Find \[ \ds \lim_{x\to 3^-} \frac{2x}{x-3}\] and vertical asymptote(s) of \( y=\frac{2x}{x-3} \).

2x/(x-3)