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nycmathguy

Member

Registered: 2021-06-02

Posts: 53

Limit of Three-piece Function

Use a graph to investigate limit of f(x) as x tends to c at the number c.

Note: f(x) is a piecewise function.

P. S. I tackled the problem algebraically not by way of graph.
When I learn how to post images, I will use more graphs throughout my study of calculus.

Top portion of piecewise function: x^2 if x < 1
Middle portion of piecewise function: 2 if x = 1
Bottom portion of piecewise function: -3x + 2 if x > 1

at c = 1

Find the limit of x^2 as x tends to 1 from the left side.

(1)^2 = 1

Find the limit of -3x + 2 as x tends to 1 from the right side.

-3(1) + 2 =

-3 + 2 = -1

LHL DOES NOT = RHL.

I say the limit of f(x) does not exist.

NOTE: I will use DNE to mean Does Not Exist moving forward in my study of limits as famously done in all calculus textbooks.

zetafunc

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Registered: 2014-05-21

Posts: 2,440

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Re: Limit of Three-piece Function

Your conclusion is correct -- however, there are a few technical points to note.

Find the limit of x^2 as x tends to 1 from the left side.

What you really want to be saying is that you are finding the limit of f(x) as x tends to 1 from the left, rather than the limit of x^2.

(1)^2 = 1

Strictly speaking, this is how you evaluate the limit of f(x) as x approaches 1 from the left, but this isn't how you evaluate the left-hand limit of x^2, this is just substituting in x = 1 -- although you can get away with it here. (This is because x^2, 2 and -3x + 2 are all continuous functions, so the left and right-hand limits for each function exist, are equal, and are equivalent to their usual limits.) The way you ought to be thinking about it is:

It'll lead to the same answer, but conceptually this is along the sort of lines you'll be required to think through when you look at the epsilon-delta type questions.

However, the question did ask for you to use a graph. So what do you think f(x) might look like on a graph? What kind of shape does it have as you move from left to right? And in particular what happens near x = 1?

nycmathguy

Member

Registered: 2021-06-02

Posts: 53

Re: Limit of Three-piece Function

Your conclusion is correct -- however, there are a few technical points to note.

Find the limit of x^2 as x tends to 1 from the left side.

What you really want to be saying is that you are finding the limit of f(x) as x tends to 1 from the left, rather than the limit of x^2.

(1)^2 = 1

Strictly speaking, this is how you evaluate the limit of f(x) as x approaches 1 from the left, but this isn't how you evaluate the left-hand limit of x^2, this is just substituting in x = 1 -- although you can get away with it here. (This is because x^2, 2 and -3x + 2 are all continuous functions, so the left and right-hand limits for each function exist, are equal, and are equivalent to their usual limits.) The way you ought to be thinking about it is:

It'll lead to the same answer, but conceptually this is along the sort of lines you'll be required to think through when you look at the epsilon-delta type questions.

However, the question did ask for you to use a graph. So what do you think f(x) might look like on a graph? What kind of shape does it have as you move from left to right? And in particular what happens near x = 1?

1. Let's not get too technical here.

2. I am not thinking about graphing piecewise functions at this early stage in my self-study of calculus.

3. As far as the delta/epsilon definition of a limit, I may or may not include this in my self-study of this course.

4. I am not a classroom student. My college days ended in 1994.

5. Why did you feel a need to bring delta into this reply? I don't know what you did here. I have not taken calculus in a classroom setting and at my age, I never will.