Piecewise Functions and limits

[Thinker1920]

Hi Everyone,

I've been trying to help my cousin with his Calc, and niether of us can figure out how his teacher got the answer for this one problem. Here's a pic.



the teacher got X -> 5+, y -> sqrt 3
X -> 5-, y -> sqrt 2
X -> 5, y -> sqrt 8

I think I understand that the two sided limit, since approaching 5 would be on line sqrt X+3, so sqrt 5+3 would be sqrt 8 if I'm understanding it right, but I don't understand how she came up with the right and left side limits. help?

 

[Ishuda]

Hi Everyone,

I've been trying to help my cousin with his Calc, and niether of us can figure out how his teacher got the answer for this one problem. Here's a pic.



the teacher got X -> 5+, y -> sqrt 3
X -> 5-, y -> sqrt 2
X -> 5, y -> sqrt 8

I think I understand that the two sided limit, since approaching 5 would be on line sqrt X+3, so sqrt 5+3 would be sqrt 8 if I'm understanding it right, but I don't understand how she came up with the right and left side limits. help?

Let's rewrite your equation and put it in order on the number line, smallest first, and also be a little redundant
p(x)=
(a) x² - 1; x \(\displaystyle \le\) -3
(b) \(\displaystyle \sqrt{x + 3}\); -3 < x < 5
(c) \(\displaystyle \sqrt{x - 5}\); x = 5
(d) \(\displaystyle \sqrt{x + 3}\); 5 < x

Now, starting at th end and working backwards:
As x -> 5+, x>5 and we use (d) to get p(x) -> \(\displaystyle \sqrt{8}\).
At x = 5, we use (c) to get p(5) = 0.
As x -> 5-, -3 < x < 5 and we use (b) to get p(x) -> \(\displaystyle \sqrt{8}\).
As x -> -3+, -3 < x < 5 and we use (b) to get p(x) -> \(\displaystyle \sqrt{0}\) = 0.
As x -> -3-, x < -3 and we use (a) to get p(x) -> 8.

 

[Thinker1920]

that makes sense but doesn't match what the teacher said is the correct answers

 

[stapel]

I think your teacher is wrong.

 

[Steven G]

Hi Everyone,

I've been trying to help my cousin with his Calc, and niether of us can figure out how his teacher got the answer for this one problem. Here's a pic.



the teacher got X -> 5+, y -> sqrt 3
X -> 5-, y -> sqrt 2
X -> 5, y -> sqrt 8

I think I understand that the two sided limit, since approaching 5 would be on line sqrt X+3, so sqrt 5+3 would be sqrt 8 if I'm understanding it right, but I don't understand how she came up with the right and left side limits. help?

Did the teacher really say sqrt(3)?

 

[Ishuda]

that makes sense but doesn't match what the teacher said is the correct answers

I do believe my answers are correct and if they differ from what the teacher says, then I believe the teacher is incorrect.