Limits of 2 functions

[w126]

The graph's formatting was bizarre, so I had to draw on lines.
Can someone please help me with numbers 1 and number 2? I keep getting wrong answers the key says -1 for both. What does the open and closed dots mean if they are separated? Thank you!

 

[R.M.]

Think of "open" like a version of < or >. You'll approach that point from one direction, but you'll never get there.
Think of "closed" like a version of $\ge$ or $\le$. You'll approach that point from one direction, and you can arrive there.

 

[Dr.Peterson]

The graph's formatting was bizarre, so I had to draw on lines.
Can someone please help me with numbers 1 and number 2? I keep getting wrong answers the key says -1 for both. What does the open and closed dots mean if they are separated? Thank you!View attachment 33879

Please show the original graph; what about it do you consider bizarre? Is it just that you wanted to emphasize the axes?

For the graph of f, is there a horizontal line where I drew one below, or does the function have no value between x=1 and 3?



For g, I'm more certain there is such a line:



In finding limits, you can ignore the dots, and think only about what happens as x approaches 1 from either side. For example, f(x) approaches different values from each side, as does g(x); but think about that their sum does. From the left, do you see that f(x) approaches 0, and g(x) approaches -1? What will their sum approach? Then look from the right side.

 

[w126]

Please show the original graph; what about it do you consider bizarre? Is it just that you wanted to emphasize the axes?

For the graph of f, is there a horizontal line where I drew one below, or does the function have no value between x=1 and 3?

View attachment 33881

For g, I'm more certain there is such a line:

View attachment 33882

In finding limits, you can ignore the dots, and think only about what happens as x approaches 1 from either side. For example, f(x) approaches different values from each side, as does g(x); but think about that their sum does. From the left, do you see that f(x) approaches 0, and g(x) approaches -1? What will their sum approach? Then look from the right side.

Wow, I completely didn't see the lines connecting the dots. Thank you for pointing that out! I understand it now.

 

[Steven G]

Wow, I completely didn't see the lines connecting the dots. Thank you for pointing that out! I understand it now.

You did not need to see those two lines to do part b and c. What happened there?
What are your answers?

What does the open and closed dots mean if they are separated? Thank you!View attachment 33879

We now know that there was a line between the open and closed circle. However, when you did not see those lines, you asked what does the open and closed dots means.
I will answer that for you. The closed dot (assuming that there was no line) means that this 'random' point is included in the graph.
The oped dot means that this 'random' point is not part of the graph. It is unusual to see an open dot that is not the end or the beginning of a curve.

 

[Dr.Peterson]

It is unusual to see an open dot that is not the end or the beginning of a curve.

That's an understatement! It is unheard-of, because it would be meaningless. "g(3) = -1; but it especially isn't -2! Be sure not to miss that!"

Luckily, that's the place where the line was made heavily enough that it's harder to miss.

Now, if the line from 1 to 3 for f were actually not there, then f would be undefined there, so f(x)+g(x) would only have a limit from the left as you approach 1.

Part (b), as you say, should never have had an issue.

Can someone please help me with numbers 1 and number 2? I keep getting wrong answers the key says -1 for both.

What difficulty did you have with (b)? What answer did you get? (My guess is that (a) confused you enough that you didn't seriously try the rest.)