Need help on piecewise function

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Define Darkness

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Oct 27, 2021
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Good evening,

I'm currently working through some homework and am utterly stuck on this question regarding piecewise functions.
Piecewise Function Question.png
I got as far as simplifying the top equation, but I'm unsure where to go from there.
 
What are the requirements for f(x) to be continuous at x=a? You can't play the game unless you know the rules!
1a) limit as x approaches a from the left of f(x) exists.
1b) limit as x approaches a from the right of f(x) exists.
2) The limit from 1a and 1b must be equal.
3) f(a) must equal this common limit from 2.

Now you need to find c and d such that f(x) is continuous at x=5.
 
lev888 said:
Can you outline how you are planning to solve it?
Click to expand...
My game plan for this question was as Jomo said, equate statement 1a to 1b, but I'd like to try and simplify the equations as much as possible before doing so, since it seems like a lot to work through simply going at the problem with the statements given.
 
Define Darkness said:
My game plan for this question was as Jomo said, equate statement 1a to 1b, but I'd like to try and simplify the equations as much as possible before doing so, since it seems like a lot to work through simply going at the problem with the statements given.
Click to expand...
I don't know about simplification, but the important part in Jomo's post is the existence of limits and the limits being equal (not the expressions).
 
Ok, well, after doing some calculations and using the squeeze method I arrived at an answer of c = 54/49 and d = 24/49.
I did so by finding two equations, 4c - 9d and d = 6 - 5c, plugging equation 2 into equation 1, solving for c, and then plugging c back into equation 2.

Would anyone be able to tell me if this is correct or not?
 
1a) As x approaches 5 from the left, does the limit of f(x) exist, and, if it does, what does it equal?
1b) As x approaches 5 from the right, does the limit of f(x) exist, and, if it does, what does it equal?
2) Are the limits from 1a and 1b equal to one another?
3) Does f(5) equal the common limit from 2?
 
Define Darkness said:
Ok, well, after doing some calculations and using the squeeze method I arrived at an answer of c = 54/49 and d = 24/49.
I did so by finding two equations, 4c - 9d and d = 6 - 5c, plugging equation 2 into equation 1, solving for c, and then plugging c back into equation 2.

Would anyone be able to tell me if this is correct or not?
Click to expand...
"4c- 9d" is NOT an equation!
 
For x< 5, \(\displaystyle \frac{x^2- 25}{x- 5}= \frac{(x- 5)(x+ 5)}{x- 5}= x+ 5\) so the limit, as x goes to 5 from below is 5+ 5= 10.
The limit, as x goes to 5 from below, of f is 10d+ c.
Cosine is always between -1 and 1 and x- 5 goes to 0 as x goes to 0 so the limit, as x goes to 5 from above, of this function is 0+ 6= 6.

In order that the limit exist we must have 10d+ c= 6.
The value of the function at x= 5 is 5c+ d.
(It is a bit peculiar to write "cx+ d when x= 5" rather than simply "5c+ d".)
In order that the limit equal the value of the function we must have 5c+ d= 6.

Solve the two equations 10d+ c= 6 and 5c+ d= 6.
 
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