Limits and Continuity

KarlieWarliee

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How do you sketch a graph that is not continuous at x=1 and yet has a limit that exists at x=1?

This was on a test and I'm doing corrections, what did I do wrong?

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How do you sketch a graph that is not continuous at x=1 and yet has a limit that exists at x=1?

This was on a test and I'm doing corrections, what did I do wrong?

attachment.php

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

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How do you sketch a graph that is not continuous at x=1 and yet has a limit that exists at x=1?

This was on a test and I'm doing corrections, what did I do wrong?

attachment.php
Assuming the image is your work, what you have is (what appears to be) a continuous graph and the problem states "How do you sketch a graph that is not continuous at x=1 and yet has a limit that exists at x=1?" This appears to be one of those sneaky problems where, although there are limits at x=1, the left and right hand limits are not the same.

With that in mind, can you now do the problem?

Also remember that the Domain is the values x can take on. Your graph shows a whole lot of x values [all of them between ?? and ??]
 
Last edited:
How do you sketch a graph that is not continuous at x=1 and yet has a limit that exists at x=1?

This was on a test and I'm doing corrections, what did I do wrong?

attachment.php
Your graph is correct. But your description of the function is completely wrong.

\(\displaystyle f(x)=\begin{cases}0 & x=1\\-x+2 &x\ne 1\end{cases}\), domain all real numbers.
 
Your graph is correct. But your description of the function is completely wrong.

\(\displaystyle f(x)=\begin{cases}0 & x=1\\-x+2 &x\ne 1\end{cases}\), domain all real numbers.
Come to think of it, I have seen the circle [as shown at (1,1)] to represent a 'repairable' non-continuous function where left and right limits are the same.
 
How do you sketch a graph that is not continuous at x=1 and yet has a limit that exists at x=1?

This was on a test and I'm doing corrections, what did I do wrong?

Code:
graph:
     ^ y
     |
   \ |
     2
     | \  y = -x + 2
     |   \
     |    *\
     |       \
-----*----1----2---> x
     |           \

...defined for all x EXCEPT x = 1.
The function was supposed to be defined for all x, so eliminating the value x = 1 from the domain violated that part. Otherwise, your function would have been fine. You just needed to add a different y-value for when x = 1. ;)
 
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