In Exercises 71 - 74 , complete the following tables and...

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be. $$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$ $$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$ $$f ( x ) = \sin \frac { 1 } { x }$$

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.
$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$
$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$
$$f ( x ) = \sin \frac { 1 } { x }$$

Key Concepts

Limit of a Function

The concept of a limit involves analyzing how a function behaves as its input approaches a particular value. It is a fundamental idea in calculus that helps define continuity, derivatives, and integrals by focusing on the value that function outputs appear to approach, even if the function does not actually reach that value.

Oscillatory Behavior

Oscillatory behavior refers to a situation where a function repeatedly varies between different values as the input approaches a certain point. This type of behavior is critical in understanding why some functions do not have a definite limit at that point, even though the function may be bounded.

Non-Existence of a Limit

A limit does not exist when a function fails to approach a unique, single value as the input nears a particular point. This non-existence typically occurs due to factors such as oscillations or divergent behavior, and it is an important consideration when evaluating the continuity and overall behavior of functions.

Transcript

00:01 Hi there.

00:02 So in exercise 72, it asks you to complete the following tables and state what you believe limited as x approaches zero of f of x to be.

00:11 So to make this much simpler, and instead of just trying to find for each individual one, this is how we can simplify it.

00:19 So for step one, you can pause the screen and follow it, but i'm going to explain it a little bit.

00:24 So step one, you want to hit y equals in the top upper left of your graph, your calculator, and then you, you can use.

00:31 You want to put in y is equal to sign of one divided by x.

00:35 So now you want to hit second window, which is the button that's to the right of y equals, and then go all the way down to independent and highlight ask.

00:48 And we're going to hit enter this time, right? so then now you're going to do second graph, which is all the way to the right most button.

00:58 And now you're going to have a blank x and y one table.

01:04 So now i'm going to start plugging in values, and then you're going to hit enter to get f of x and enter more x values.

01:11 So this is just for those who didn't know how to do this method in your graphing calculator.

01:17 So now we can fill out the table a lot easier instead of having, you know, to plug in each and an individual one.

01:24 So now in the x equals you want to put in negative, 0 .1 and we get the value 0 .544 about and then now we want to put in negative 0 .01 and you're going to get 0 .506 negative 0 .001.

01:49 You're going to get negative 0 .801.

01:49 You're going to get negative 0 .827 about.

01:55 Negative 0 .001.

01:58 You're going to get negative 0 .007 about negative 0 .00001.

01:59 0 .306.

02:03 Let's repeat for the rest of the values...

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