MA441-Lecture 2

MA441- LEC 2 - The limit of a function I

Goal

Understanding the definition of function
Understanding the verval definition of the limit of a function
Making a table for a function
importance of the graph for the limit. (Table is not enough.)

function

Function is a relation between a set of inputs (X) and a set of possible outputs (Y) with the following property.

For each x in X, there exists the unique y in Y

Therefore, the limit of a function is about the relation between x's and y's.

Tangent line of \(y=\frac{x^2}{4}\) at \( x_0=2\)

The slope of secant line of \( (x_0,f(x_0))\) and \((x_0+\Delta x,f(x_0+\Delta x))\) is given by

\[ \frac{\Delta y}{\Delta x}=\frac{f(x_0 + \Delta x) - f(x_0)}{x_0+\Delta x-x_0}=\frac{f(\Delta x+x_0)-f(x_0)}{\Delta x} \]

For example, if \(\Delta x=1\), then

\[ \frac{\Delta y}{\Delta x}=\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x} \]

\[ =\frac{f(2+1)-f(2)}{1}=\frac{2.25-1}{1}=1.25 \]

table of a slope of secant line

\( \Delta x \)	1	0.5	0.01	0.001	0.0001	-0.0001
slope of secant line	1.25	1.125	1.0025	1.0025	1.00003	0.99998

From the table, we can guess (but we cannot confirm yet),

\[ 0.99998 \leq \lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x} \leq 1.00003 \]

graph.

Conclusion

From the table and the graph, we expect that

\[ 1=\lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}=the~slope~of~tagent~line~at~x_0=2 \]

It means we can make the values of \(\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}\) arbitrarily to close to a certain number by choosing very small \(\Delta x \neq 0\).

Example 1

Let \( f(x)=x^2 \). Then \(\lim_{x\to 2} f(x)=?\)

result

From the graph, we can conclude

\[ \lim_{x\to 2} f(x)=4 \]

definition of the limit of a function \(y=f(x)\) at \(x=x_0\).

Suppose f(x) defined when x is near the number a except \(x=x_0\). (This means that f is defined on some open interval that contains \(x_0\), except possibly at \(x_0\) itself.) Then we write

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

and say "the limit of f(x), as x approaches \(x_0\), equals L".

if we make the values of f(x) arbitrarily close L by taking all x's to be sufficiently close to \(x_0\) but not equal to \(x_0\).

Facts

Do not forget, \(x\neq x_0\).
\(f(x_0) = \lim_{x\to x_0}f(x) \)is not true in general.
Actually, \(f(x_0)\) does not need to be defined to define the limit of \(f(x)\) at \(x=x_0\).

3 basic cases

1. Continuity	2. Hole and \(f(x_0)\) is defined.	3. Hole and \(f(x_0)\) is undefined.

Example 2

Let \(f(x)=\frac{x^2-4}{x-2}\). \(\lim_{x\to 2}f(x)=?\)

Example 3. importance of graph with \(y=\sin(\frac{\pi}{x})\)

Make the table.

Let \(x=\frac{1}{n}\) where \(n\) is a natural number. Then

\[f(\frac{1}{n})=\sin(n\pi)=0 \]

So we may guess that \[\lim_{x\to 0} \sin(\frac{\pi}{x})=0\]

But it is wrong. Actually, there is no limit at \(x=0\).