Find the constant k that will make this piecewise continuous.


by kaderyo94
Tags: constant, continuous, piecewise
kaderyo94
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#1
Oct2-12, 10:34 AM
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1. The problem statement, all variables and given/known data
Find a value for the constant k that will make the function below continuous:

[itex]f(x)=\frac{x-1}{x^2-1}\ \text{if}\ x<=0[/itex]
[itex]f(x)=\frac{tankx}{2x}~\text{if}~x>0[/itex]



2. Relevant equations



3. The attempt at a solution
I've tried the only solution I can think of, which is to make
[itex]\frac{x-1}{x^2-1} = \frac{tankx}{2x}[/itex]

And then I plug in 0 to try and get k, but I end up with 1 = 0/0. I know you are supposed to do something to the second equation to remedy this, but I cannot figure out what. I am fairly new to Calculus, so some help would be greatly appreciated.
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SammyS
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Oct2-12, 11:36 AM
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Quote Quote by kaderyo94 View Post
1. The problem statement, all variables and given/known data
Find a value for the constant k that will make the function below continuous:

[itex]f(x)=\frac{x-1}{x^2-1}\ \text{if}\ x<=0[/itex]

[itex]f(x)=\frac{\tan kx}{2x}~\text{if}~x>0[/itex]

2. Relevant equations

3. The attempt at a solution
I've tried the only solution I can think of, which is to make
[itex]\frac{x-1}{x^2-1} = \frac{\tan kx}{2x}[/itex]

And then I plug in 0 to try and get k, but I end up with 1 = 0/0. I know you are supposed to do something to the second equation to remedy this, but I cannot figure out what. I am fairly new to Calculus, so some help would be greatly appreciated.
Hello kaderyo94. Welcome to PF !

Each piece of this piecewise-defined function has one or more discontinuities in its portion of the domain of the overall function. You can't "fix" those discontinuities by a choice of k.

I suspect the problem is: Find a value for the constant k that will make the function continuous at x = 0, which is where the two "pieces" join.

If that's the problem to be solved, then:

What must be true for the following limit to exist?
[itex]\displaystyle \lim_{x\to\,0}\,f(x)[/itex]
Then, how must that limit be related to f(0) ?
HallsofIvy
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#3
Oct2-12, 05:22 PM
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You titled this "Find the constant k that will make this piecewise continuous" but then said "Find a value for the constant k that will make this function continous". Those are very different!


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