Use rationalization to simplify the given expression in part (a). Then find the indicated limit in part (b).
(a.)
3y2/(the square root of (y2 + y + 1))−(the square root of (y + 1))
b.)lim y>0 3y2/(the square root of (y2 + y + 1))−(the square root of (y + 1))
Do you mean
[3y^2/√(y^2+y+1)] - √(y+1)
or
3y^2 / [√(y^2+y+1) - √(y+1)]
?
the first one
no I mean the second one
3y^2 / [�ã(y^2+y+1) - �ã(y+1)] sorry
I thought so, but thought I'd check.
1/[√(y^2+y+1) - √(y+1)]
multiply top and bottom by
√(y^2+y+1) + √(y+1) and you have
[√(y^2+y+1) + √(y+1)]/(y^2+y+1 - (y+1))
= [√(y^2+y+1) + √(y+1)]/(y^2-1)
So now, a couple of applications of l'Hospital's Rule will get rid of that pesky 0/0.
To simplify the given expression in part (a), we can use rationalization. Rationalizing the denominator involves getting rid of any square roots in the denominator by multiplying the expression by the conjugate of the denominator.
(a.)
To rationalize the expression, we multiply the numerator and denominator of the fraction by the conjugate of the denominator, which is the square root of (y2 + y + 1) + the square root of (y + 1).
So, the expression becomes:
(3y2/(the square root of (y2 + y + 1))−(the square root of (y + 1))) * ((the square root of (y2 + y + 1)) + (the square root of (y + 1)))/((the square root of (y2 + y + 1)) + (the square root of (y + 1)))
Expanding the numerator and denominator, we get:
(3y2 * (the square root of (y2 + y + 1))) + (3y2 * (the square root of (y + 1))) - (3y2 * (the square root of (y + 1))) - ((the square root of (y2 + y + 1))^2 - (the square root of (y + 1))^2)
Simplifying further, we get:
(3y2 * (the square root of (y2 + y + 1))) / ((y2 + y + 1) - (y + 1))
= (3y2 * (the square root of (y2 + y + 1))) / (y2 + y - y - 1)
= (3y2 * (the square root of (y2 + y + 1))) / (y2 - 1)
Now, we move on to part (b) where we need to find the limit.
b.)
To find the limit of the expression as y approaches positive infinity, we can simplify the expression and evaluate it directly.
Substituting in the simplified expression from part (a):
lim y->0 (3y2 * (the square root of (y2 + y + 1))) / (y2 - 1)
As y approaches positive infinity, the denominator becomes positive infinity, and the numerator also becomes positive infinity.
Therefore, the limit of the expression as y approaches positive infinity is infinity.