MA441-Lecture 3

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)

MA441- LEC 3 - The limit of a function II

Goal

Example 1. (Jump)

Left-handed limit

right-handed limit

proposition.

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

Check up

infinite limits

Example.

Example.

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

Example.