Find the values of a and b that make f continuous everywhere

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In summary, the values of a and b that make f continuous everywhere are a=1/8b and b=-28/11. The first step is to factor the numerator and cancel the factor which is going to zero. Then, use algebra to solve for a and b by setting up and solving equations. It is important to take breaks and not stare at a problem for too long to avoid making mistakes.
  • #1
badatmaths
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Homework Statement


Find the values of a and b that make f continuous everywhere. (Enter your answers as fractions.)


Homework Equations


1mLJ9.gif



The Attempt at a Solution



lim x->2 (x2-4)/(x-2) = 0

lim x->2 4a - 2b +5
lim x->3 9a - 3b +5

lim x->3 12 - a + b


4a - 2b + 5 = 0
4a - 2b = -5

9a - 3b +5 = 12 - a + b
10a - 4b = 7


4a - 2b = -5
a = 2.5b/4

10(2.5b/4) - 4b = 7
6.25b - 4b = 7
2.25b = 7
b = 28/9


I'm obviously missing something because I've been screwing around with this problem for well over an hour already and I haven't been able to solve it. I need to get this done by tomorrow morning so any help is appreciated.
 

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  • #2
You've got the right method, but there's a problem with the first step. lim x->2 (x^2-4)/(x-2) isn't zero.
 
  • #3
Dick said:
You've got the right method, but there's a problem with the first step. lim x->2 (x^2-4)/(x-2) isn't zero.

I'm really not sure what to do if x->2 for the first equation is undefined...
 
  • #4
badatmaths said:
I'm really not sure what to do if x->2 for the first equation is undefined...

Try and factor the numerator and cancel the factor which is going to zero.
 
  • #5
Dick said:
Try and factor the numerator and cancel the factor which is going to zero.

(x+2)(x-2)/(x-2)
(x+2)
x->2 = 4

That would mean that...

4a - 2b + 5 = 4
4a - 2b = -1
4a = 1/2b
a = 1/8b

But plugging that into 10a - 4b = 7 gives me...

10(1/8b) - 4b = 7
(5/4)b - 4b = 7
(-11/4)b = 7
b = -28/11

...Okay, that isn't right...I did this problem 3 different times and came up with a new answer each time.
 
  • #6
badatmaths said:
(x+2)(x-2)/(x-2)
(x+2)
x->2 = 4

That would mean that...

4a - 2b + 5 = 4
4a - 2b = -1
4a = 1/2b
a = 1/8b

But plugging that into 10a - 4b = 7 gives me...

10(1/8b) - 4b = 7
(5/4)b - 4b = 7
(-11/4)b = 7
b = -28/11

...Okay, that isn't right...I did this problem 3 different times and came up with a new answer each time.

You are goofing up your algebra. 4a-2b=(-1) doesn't lead to 4a=(1/2)b. What are you doing?
 
  • #7
Dick said:
You are goofing up your algebra. 4a-2b=(-1) doesn't lead to 4a=(1/2)b. What are you doing?

I'm going to blame it on the embarrassingly long amount of time I've been looking at this problem. I just solved, finally. Thanks a lot! I really appreciate it!
 
  • #8
badatmaths said:
I'm going to blame it on the embarrassingly long amount of time I've been looking at this problem. I just solved, finally. Thanks a lot! I really appreciate it!

Staring at a problem for too long does make this stuff happen. Glad you got it.
 

FAQ: Find the values of a and b that make f continuous everywhere

What does it mean for a function to be continuous everywhere?

For a function to be continuous everywhere, it must have no breaks or holes in its graph and must be defined at every point in its domain. This means that the function must have a smooth and continuous curve without any sharp corners or jumps.

How do I know if a function is continuous everywhere?

A function can be checked for continuity everywhere by looking at its graph and ensuring that there are no breaks or gaps. Additionally, the function must be defined at every point in its domain. If there are any points where the function is not defined or has a jump or discontinuity in its graph, then it is not continuous everywhere.

Why is it important to find the values of a and b that make f continuous everywhere?

Finding the values of a and b that make f continuous everywhere is important because it ensures that the function is well-behaved and can be used to make accurate predictions and calculations. It also allows us to better understand the behavior of the function and its graph.

How do I find the values of a and b that make f continuous everywhere?

To find the values of a and b that make f continuous everywhere, we can use the limit definition of continuity. This involves taking the limit of the function as it approaches a specific point and setting it equal to the value of the function at that point. By solving for a and b in this equation, we can determine the values that make the function continuous everywhere.

Can a function be continuous everywhere without having a specific value for a and b?

Yes, it is possible for a function to be continuous everywhere without having specific values for a and b. This can occur when the function is already defined as continuous and does not require any additional parameters to make it continuous everywhere. However, in most cases, finding the values of a and b is necessary to ensure continuity everywhere.

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