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nycmathguy

Member

Registered: 2021-06-02

Posts: 53

Limit of Piecewise Functions

Investigate the limit of f(x) as x tends to c at the number c.

{2x + 1 if x ≤ 0....top portion of piecewise function.
{2x if x > 0..........bottom portion of piecewise function.

The number c = 0.

Solution:

If c = 0, I should use the top portion of f(x).

That is, 2x + 1.

Let x = 0.

2(0) + 1 = 0 + 1 = 1.

I say the limit of f(x) is 0 when c = 0.

I hope this is correct.

Last edited by nycmathguy (2021-06-03 17:32:28)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Limit of Piecewise Functions

Is there a bit missing from this question? The definitions for the function do not have any 'c' in them, so how are we supposed to know what c is.  Or has your finger just slipped and typed c when you meant x?

I've sketched the graph and there's a discontinuity at x = 0. The left limit is y=1 and the right limit is y = 0.

So the usual conclusion is to say there is no consistent limit at x = 0.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycmathguy

Member

Registered: 2021-06-02

Posts: 53

Re: Limit of Piecewise Functions

Is there a bit missing from this question? The definitions for the function do not have any 'c' in them, so how are we supposed to know what c is.  Or has your finger just slipped and typed c when you meant x?

I've sketched the graph and there's a discontinuity at x = 0. The left limit is y=1 and the right limit is y = 0.

So the usual conclusion is to say there is no consistent limit at x = 0.

Bob

Hello Bob. Good morning. I did not make a typo.

Here are the specific instructions:

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

Sullivan provided the value of c to be 0. I decided to solve it algebraically. However, if I must check the limit from the left and right, how is this done algebraically in terms of piecewise functions?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Limit of Piecewise Functions

OK. I'll say what I think is the answer.

I think that all values of c need to be considered.

(1) c< 0 lim = 2c+1

(2) c> 0 lim = 2c

(c) c=0, indeterminate.

That's the best I can offer I'm afraid.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycmathguy

Member

Registered: 2021-06-02

Posts: 53

Re: Limit of Piecewise Functions

OK. I'll say what I think is the answer.

I think that all values of c need to be considered.

(1) c< 0 lim = 2c+1

(2) c> 0 lim = 2c

(c) c=0, indeterminate.

That's the best I can offer I'm afraid.

Bob

Unfortunately, this is an even number problem in the book to which there is no answer in the back section. I don't know if you are right or wrong.

nycmathguy

Member

Registered: 2021-06-02

Posts: 53

Re: Limit of Piecewise Functions

OK. I'll say what I think is the answer.

I think that all values of c need to be considered.

(1) c< 0 lim = 2c+1

(2) c> 0 lim = 2c

(c) c=0, indeterminate.

That's the best I can offer I'm afraid.

Bob

Unfortunately, this is an even number problem in the book to which there is no answer in the back section. I don't know if you are right or wrong.

I went back to see if the author has a sample problem for piecewise functions.
Luckily, Sullivan does provide a sample for a similar question. I will use the sample question in the textbook as a guide to help me solve problem 30.

30. Investigate the limit of f(x) as x tends to c at the number c.

The number c is given to be 0.

{2x + 1 if x ≤ 0....top portion of piecewise function.
{2x if x > 0..........bottom portion of piecewise function.

We want to do find the limit as x tends to 0 from the left and right.

Find the limit (2x + 1) as x tends to 0 from the left.

2(0) + 1 = 1

Find the limit of 2x as x tends to 0 from the right.

2(0) = 0

The left handed limit DOES NOT EQUAL the right handed limit.

Thus, the limit of f(x) as x tends 0 does not exist.

You say?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Limit of Piecewise Functions

That's what I said in post 2.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycmathguy

Member

Registered: 2021-06-02

Posts: 53

Re: Limit of Piecewise Functions

That's what I said in post 2.

Bob

I like calculus already. I will post a few more piecewise functions before moving on.
I do hope my questions and replies help others as well.