Use the graph of f(x) to investigate the limit

In summary, the graph of f(x) is used to investigate the limit of the piecewise-defined function as x tends to c at the number c. After examining the graph and calculating the left and right one-sided limits, it is determined that the limit does not exist. This conclusion holds true regardless of whether the function includes an "=" sign in its definition. Well done on solving this problem.
  • #1
nycmathguy
Homework Statement
Use the graph of f(x) to investigate the limit.
Relevant Equations
Piecewise-defined Function
Use the graph to investigate the limit of f(x) as x tends to c at the number c.

See attachments.

Based on the graph of f(x), here is what I did:

lim (2x + 1) as x tends to 0 from the left is 1.

lim (2x) as x tends to 0 from the right is 0.

LHL does not equal RHL.

I conclude the limit of f(x) as x tends to c at the number c (c = 0) does not exist.
 

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  • #2
nycmathguy said:
Homework Statement:: Use the graph of f(x) to investigate the limit.
Relevant Equations:: Piecewise-defined Function

I conclude the limit of f(x) as x tends to c at the number c (c = 0) does not exist.
Um, one of your piecewise continuous definitions includes and "=" sign...
 
  • #3
nycmathguy said:
I conclude the limit of f(x) as x tends to c at the number c (c = 0) does not exist.
Looks good to me.
 
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  • #4
berkeman said:
Um, one of your piecewise continuous definitions includes and "=" sign...
Meaning?
 
  • #5
Mark44 said:
Looks good to me.

That's what I thought. I may post one more like this involving a piecewise function in terms of 3 pieces. Let me think about it.
 
  • #6
berkeman said:
Um, one of your piecewise continuous definitions includes and "=" sign...
nycmathguy said:
Meaning?
Doesn't matter whether one or both or neither of the function definitions has an = in it. The one-sided limits you found are correct, and your conclusion that the two-sided limit doesn't exist is also correct.

Well done.
 
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  • #7
Mark44 said:
Doesn't matter whether one or both or neither of the function definitions has an = in it. The one-sided limits you found are correct, and your conclusion that the two-sided limit doesn't exist is also correct.

Well done.

Beautiful. I may post one more but this time, the piecewise function will be three pieces. Stay tune.
 

FAQ: Use the graph of f(x) to investigate the limit

What is a limit?

A limit is a mathematical concept that describes the behavior of a function as the input (x) approaches a certain value. It represents the value that the function is "approaching" as x gets closer and closer to the given value.

How do I find the limit of a function?

To find the limit of a function, you can use algebraic techniques such as factoring, simplifying, or using special limits. You can also use graphical methods by looking at the behavior of the function on a graph.

What does the graph of f(x) tell us about the limit?

The graph of f(x) can give us a visual representation of the behavior of the function as x approaches a certain value. It can show us if the limit exists, if it is finite or infinite, and if there are any discontinuities or holes in the function.

How can I use the graph to investigate the limit?

You can use the graph to estimate the limit by looking at the behavior of the function as x gets closer and closer to the given value. You can also use the graph to identify any potential issues or special cases, such as vertical asymptotes or removable discontinuities.

What are some common mistakes when investigating the limit using a graph?

Some common mistakes when investigating the limit using a graph include not considering the behavior of the function on both sides of the given value, not taking into account any special cases or discontinuities, and not using appropriate notation to indicate the limit.

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