lim y^3-y^2-y-2/2y^3-5y^2+5y-6
y-->2
To find the limit of the expression (y^3 - y^2 - y - 2) / (2y^3 - 5y^2 + 5y - 6) as y approaches 2, we can substitute y = 2 into the expression and simplify:
Replace y with 2 in the expression:
(2^3 - 2^2 - 2 - 2) / (2(2^3) - 5(2^2) + 5(2) - 6)
Simplify the expression:
(8 - 4 - 2 - 2) / (16 - 20 + 10 - 6)
(0) / (0)
The expression becomes undefined because we end up with a division by zero. In this case, we cannot determine the limit as y approaches 2 as the expression does not exist.
To find the limit of a function as y approaches a specific value, we can substitute that value into the function and simplify. Here's how we can find the limit of the given function as y tends to 2:
1. Substitute 2 for y in the function:
(2^3 - 2^2 - 2 - 2) / (2(2^3) - 5(2^2) + 5(2) - 6)
2. Simplify the numerator and denominator:
(8 - 4 - 2 - 2) / (16 - 20 + 10 - 6)
(0) / (0)
At this point, we have obtained an indeterminate form of 0/0, which means we cannot determine the limit just by substituting the value. We need to try a different approach.
3. Factorize both the numerator and denominator:
Numerator: y^3 - y^2 - y - 2 = (y - 2)(y^2 + y + 1)
Denominator: 2y^3 - 5y^2 + 5y - 6 = (y - 2)(2y^2 + 4y + 3)
4. Cancel out common factors if any:
(y - 2)(y^2 + y + 1) / (y - 2)(2y^2 + 4y + 3)
5. Simplify further:
(y^2 + y + 1) / (2y^2 + 4y + 3)
6. Now substitute 2 for y in the simplified expression:
(2^2 + 2 + 1) / (2(2^2) + 4(2) + 3)
(4 + 2 + 1) / (8 + 8 + 3)
(7) / (19)
Therefore, as y approaches 2, the limit of the given function is 7/19.
Have you tried these yourself? For the first couple you just plug in the value of the variable. No limit process is necessary.
for this one
Now I bet that y-2 is a factor of both top and bottom or the denominator would not go to zero causing our problem to be non trivial at y = 2
So try dividing top and bottom by (y-2)
for the numerator I get y^2+y+1
for the denominator I get 2y^2-y+3
so I have at y = 2
(4+2+1)/(8-2+3)
7/9