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Calculus
Section 2.5
How is this done algebraically?
Section 2.5
How is this done algebraically?
to make f continuous, you need to find intersection points of all three given functions:
start with first function
(x^2-4)/(x-2)=x+2
second function is: ax^2-bx+3
equal it to first function:
ax^2-bx+3=x+2..........solve for a
ax^2=bx-3+x+2
ax^2=bx+x-1
a=(bx+x-1)/x^2.........eq.1
do same using third function:
2x-a+b=x+2......solve for a
2x-x+b-2=a
a=x+b-2...........eq.2
from eq.1 and eq.2 we haveL
(bx+x-1)/x^2=x+b-2 ......solve for b
b (x^2 - x) = -x^3 + 2 x^2 + x - 1
b = (-x^3 + 2 x^2 + x - 1)/(x^2 - x)
go to eq/2, substitute b
a=x+ (-x^3 + 2 x^2 + x - 1)/(x^2 - x) -2
a=(x^2 - 3 x + 1)/(x - x^2)
now we have a and be expressed in terms of x
so, second function will be
ax^2-bx+3=((x^2 - 3 x + 1)/(x - x^2))x^2-( (-x^3 + 2 x^2 + x - 1)/(x^2 - x) )x+3=x+2
=>a=0, b=-1
third function will be
2x-a+b=2x-0-1=2x-1
so,
ax^2-bx+3=x+2
2x-a+b=2x-0-1=2x-1
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is continuous![]()