Please determine the following limits if they exist. If the limit does not exist put DNE.
lim 2+6x-3x^2 / (2x+1)^2
x-> - infinity
lim 4n-3 / 3n^2+2
n-> infinity
I did
lim 2+6x-3x^2 / (2x+1)^2
x-> - infinity
(2+6x-3x²)/(4x²+4x+1)
(2/x² + 6/x -3)/(4 + 4/x + 1/x²)
as x-> - infinity. My answer is -3/4
lim 4n-3 / 3n^2+2
n-> infinity
Answer is 0.
both are right.
To find the limits, you can use the fact that as x or n approaches negative or positive infinity, the terms with the highest powers dominate the expression. Here's how you can find the limits step by step:
1. For the first limit, we have:
lim (2+6x-3x^2) / (2x+1)^2 as x approaches negative infinity.
2. Divide every term in the numerator and denominator by x^2 to simplify the expression:
lim (2/x^2 + 6/x -3)/(4 + 4/x + 1/x^2) as x approaches negative infinity.
3. As x approaches negative infinity, both 1/x and 1/x^2 approach 0. We can ignore the terms involving 1/x and 1/x^2, as they become negligible:
lim (0 + 0 - 3)/(4 + 0 + 0) as x approaches negative infinity.
4. Simplify the expression:
lim (-3)/(4) as x approaches negative infinity.
5. The limit is -3/4.
For the second limit:
1. We have:
lim (4n-3) / (3n^2+2) as n approaches infinity.
2. As n approaches infinity, the terms 4n and 3n^2 dominate the expression. Divide every term by n^2 to simplify the expression:
lim (4/n -3/n^2)/(3 + 2/n^2) as n approaches infinity.
3. As n approaches infinity, both 1/n and 1/n^2 approach 0. We can ignore the terms involving 1/n and 1/n^2, as they become negligible:
lim (0 - 0)/(3 + 0) as n approaches infinity.
4. Simplify the expression:
lim 0/3 as n approaches infinity.
5. The limit is 0.
Therefore, the limits are:
-3/4 as x approaches negative infinity
0 as n approaches infinity.